Examine individual changes
This page allows you to examine the variables generated by the Edit Filter for an individual change.
Variables generated for this change
| Variable | Value |
|---|---|
Edit count of the user (user_editcount) | null |
Name of the user account (user_name) | '112.201.170.228' |
Age of the user account (user_age) | 0 |
Groups (including implicit) the user is in (user_groups) | [
0 => '*'
] |
Rights that the user has (user_rights) | [
0 => 'createaccount',
1 => 'read',
2 => 'edit',
3 => 'createtalk',
4 => 'writeapi',
5 => 'viewmywatchlist',
6 => 'editmywatchlist',
7 => 'viewmyprivateinfo',
8 => 'editmyprivateinfo',
9 => 'editmyoptions',
10 => 'abusefilter-log-detail',
11 => 'centralauth-merge',
12 => 'abusefilter-view',
13 => 'abusefilter-log',
14 => 'vipsscaler-test'
] |
Whether the user is editing from mobile app (user_app) | false |
Whether or not a user is editing through the mobile interface (user_mobile) | false |
Page ID (page_id) | 25685 |
Page namespace (page_namespace) | 0 |
Page title without namespace (page_title) | 'Random variable' |
Full page title (page_prefixedtitle) | 'Random variable' |
Edit protection level of the page (page_restrictions_edit) | [] |
Last ten users to contribute to the page (page_recent_contributors) | [
0 => 'Monkbot',
1 => 'Michaelboerman420',
2 => 'Ladislav Mecir',
3 => 'OlliverWithDoubleL',
4 => 'Gsokolov',
5 => 'D.Lazard',
6 => '122.163.184.244',
7 => 'Tom.Reding',
8 => 'Miaumee',
9 => 'Walwal20'
] |
Page age in seconds (page_age) | 630183898 |
Action (action) | 'edit' |
Edit summary/reason (summary) | '/* Discrete random variable */ ' |
Old content model (old_content_model) | 'wikitext' |
New content model (new_content_model) | 'wikitext' |
Old page wikitext, before the edit (old_wikitext) | '{{short description|Variable representing a random phenomenon}}
{{Probability fundamentals}}
In [[probability]] and [[statistics]], a '''random variable''', '''random quantity''', '''aleatory variable''', or '''stochastic variable''' is described informally as a [[Dependent and independent variables|variable whose values depend]] on [[Outcome (probability)|outcomes]] of a [[Randomness|random]] phenomenon.<ref>{{cite book|last1=Blitzstein|first1=Joe|last2=Hwang|first2=Jessica|title=Introduction to Probability|date=2014|publisher=CRC Press|isbn=9781466575592}}</ref> The formal mathematical treatment of random variables is a topic in [[probability theory]]. In that context, a random variable is understood as a [[measurable function]] defined on a [[probability space]] that maps from the [[sample space]] to the [[real number]]s.<ref name="UCSB">{{cite web | title = Economics 245A – Introduction to Measure Theory | url = http://econ.ucsb.edu/~doug/245a/Lectures/Measure%20Theory.pdf | last = Steigerwald | first = Douglas G. | publisher = University of California, Santa Barbara | access-date = April 26, 2013}}</ref>
[[File:Random Variable as a Function-en.svg|thumb|This graph shows how random variable is a function from all possible outcomes to real values. It also shows how random variable is used for defining probability mass functions.]]
A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, because of imprecise measurements or [[quantum uncertainty]]). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but is instead related to philosophical arguments over the [[interpretation of probability]]. The mathematics works the same regardless of the particular interpretation in use.
As a function, a random variable is required to be [[Measurable function|measurable]], which allows for probabilities to be assigned to sets of its potential values. It is common that the outcomes depend on some physical variables that are not predictable. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physical conditions, so the outcome being observed is uncertain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.
The [[___domain of a function|___domain]] of a random variable is called a ''sample space,'' defined as the set of possible outcomes of a non-deterministic event. For example, in the event of a coin toss, only two possible outcomes are possible: heads or tails.
A random variable has a [[probability distribution]], which specifies the probability of [[Borel subset]]s of its range. Random variables can be [[Discrete random variable|discrete]], that is, taking any of a specified finite or [[countable set|countable list]] of values (having a countable range), endowed with a [[probability mass function]] that is characteristic of the random variable's probability distribution; or [[Continuous random variable|continuous]], taking any numerical value in an interval or collection of intervals (having an [[Uncountable set|uncountable]] range), via a [[probability density function]] that is characteristic of the random variable's probability distribution; or a mixture of both.
Two random variables with the same probability distribution can still differ in terms of their associations with, or [[independence (probability theory)|independence]] from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called [[random variate]]s.
==Definition==
A '''random variable''' is a [[measurable function]] <math>X \colon \Omega \to E</math> from a set of possible [[outcome (probability)|outcome]]s <math> \Omega </math> to a [[measurable space]] <math> E</math>. The technical axiomatic definition requires <math>\Omega</math> to be a sample space of a [[probability space|probability triple]] <math>(\Omega, \mathcal{F}, \operatorname{P})</math> (see the [[#Measure-theoretic definition|measure-theoretic definition]]). A random variable is often denoted by capital [[Latin script|roman letters]] such as <math>X</math>, <math>Y</math>, <math>Z</math>, <math>T</math>.<ref name=":1">{{Cite web|date=2020-04-26|title=List of Probability and Statistics Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/probability-statistics-symbols/|access-date=2020-08-21|website=Math Vault|language=en-US}}</ref><ref>{{Cite web|title=Random Variables|url=https://www.mathsisfun.com/data/random-variables.html|access-date=2020-08-21|website=www.mathsisfun.com}}</ref>
The probability that <math>X</math> takes on a value in a measurable set <math>S\subseteq E</math> is written as
: <math>\operatorname{P}(X \in S) = \operatorname{P}(\{ \omega \in \Omega \mid X(\omega) \in S \})</math><ref name=":1" />
===Standard case===
In many cases, <math>X</math> is [[Real number|real-valued]], i.e. <math>E = \mathbb{R}</math>. In some contexts, the term [[random element]] (see [[#Extensions|extensions]]) is used to denote a random variable not of this form.
{{Anchor|Discrete random variable}}When the [[Image (mathematics)|image]] (or range) of <math>X</math> is [[countable set|countable]], the random variable is called a '''discrete random variable'''<ref name="Yates">{{cite book | last = Yates | first = Daniel S. | last2 = Moore | first2 = David S | last3 = Starnes | first3 = Daren S. | year = 2003 | title = The Practice of Statistics | edition = 2nd | publisher = [[W. H. Freeman and Company|Freeman]] | ___location = New York | url = http://bcs.whfreeman.com/yates2e/ | isbn = 978-0-7167-4773-4 | url-status = dead | archive-url = https://web.archive.org/web/20050209001108/http://bcs.whfreeman.com/yates2e/ | archive-date = 2005-02-09 }}</ref>{{rp|399}} and its distribution is a [[discrete probability distribution]], i.e. can be described by a [[probability mass function]] that assigns a probability to each value in the image of <math>X</math>. If the image is uncountably infinite (usually an [[Interval (mathematics)|interval]]) then <math>X</math> is called a '''continuous random variable'''.<ref>{{Cite web|title=Random Variables|url=http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm|access-date=2020-08-21|website=www.stat.yale.edu}}</ref>{{Citation needed|reason=Statistics 101 is hardly a sufficient citation, even coming from Yale|date=October 2020}} In the special case that it is [[absolutely continuous]], its distribution can be described by a [[probability density function]], which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous,<ref>{{cite book|author1=L. Castañeda |author2=V. Arunachalam |author3=S. Dharmaraja |name-list-style=amp |title = Introduction to Probability and Stochastic Processes with Applications | year = 2012 | publisher= Wiley | page = 67 | url=https://books.google.com/books?id=zxXRn-Qmtk8C&pg=PA67 |isbn=9781118344941 }}</ref> a [[mixture distribution]] is one such counterexample; such random variables cannot be described by a probability density or a probability mass function.
Any random variable can be described by its [[cumulative distribution function]], which describes the probability that the random variable will be less than or equal to a certain value.
===Extensions===
The term "random variable" in statistics is traditionally limited to the [[real number|real-valued]] case (<math>E=\mathbb{R}</math>). In this case, the structure of the real numbers makes it possible to define quantities such as the [[expected value]] and [[variance]] of a random variable, its [[cumulative distribution function]], and the [[moment (mathematics)|moment]]s of its distribution.
However, the definition above is valid for any [[measurable space]] <math>E</math> of values. Thus one can consider random elements of other sets <math>E</math>, such as random [[Boolean-valued function|boolean value]]s, [[categorical variable|categorical value]]s, [[Covariance matrix#Complex random vectors|complex numbers]], [[random vector|vector]]s, [[random matrix|matrices]], [[random sequence|sequence]]s, [[Tree (graph theory)|tree]]s, [[random compact set|set]]s, [[shape]]s, [[manifold]]s, and [[random function|function]]s. One may then specifically refer to a ''random variable of [[data type|type]] <math>E</math>'', or an ''<math>E</math>-valued random variable''.
This more general concept of a [[random element]] is particularly useful in disciplines such as [[graph theory]], [[machine learning]], [[natural language processing]], and other fields in [[discrete mathematics]] and [[computer science]], where one is often interested in modeling the random variation of non-numerical [[data structure]]s. In some cases, it is nonetheless convenient to represent each element of <math>E</math>, using one or more real numbers. In this case, a random element may optionally be represented as a [[random vector|vector of real-valued random variables]] (all defined on the same underlying probability space <math>\Omega</math>, which allows the different random variables to [[mutual information|covary]]). For example:
*A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are <math>(1 \ 0 \ 0 \ 0 \ \cdots)</math>, <math>(0 \ 1 \ 0 \ 0 \ \cdots)</math>, <math>(0 \ 0 \ 1 \ 0 \ \cdots)</math> and the position of the 1 indicates the word.
*A random sentence of given length <math>N</math> may be represented as a vector of <math>N</math> random words.
*A [[random graph]] on <math>N</math> given vertices may be represented as a <math>N \times N</math> matrix of random variables, whose values specify the [[adjacency matrix]] of the random graph.
*A [[random function]] <math>F</math> may be represented as a collection of random variables <math>F(x)</math>, giving the function's values at the various points <math>x</math> in the function's ___domain. The <math>F(x)</math> are ordinary real-valued random variables provided that the function is real-valued. For example, a [[stochastic process]] is a random function of time, a [[random vector]] is a random function of some index set such as <math>1,2,\ldots, n</math>, and [[random field]] is a random function on any set (typically time, space, or a discrete set).
==Distribution functions==
If a random variable <math>X\colon \Omega \to \mathbb{R}</math> defined on the probability space <math>(\Omega, \mathcal{F}, \operatorname{P})</math> is given, we can ask questions like "How likely is it that the value of <math>X</math> is equal to 2?". This is the same as the probability of the event <math>\{ \omega : X(\omega) = 2 \}\,\! </math> which is often written as <math>P(X = 2)\,\!</math> or <math>p_X(2)</math> for short.
Recording all these probabilities of output ranges of a real-valued random variable <math>X</math> yields the [[probability distribution]] of <math>X</math>. The probability distribution "forgets" about the particular probability space used to define <math>X</math> and only records the probabilities of various values of <math>X</math>. Such a probability distribution can always be captured by its [[cumulative distribution function]]
:<math>F_X(x) = \operatorname{P}(X \le x)</math>
and sometimes also using a [[probability density function]], <math>p_X</math>. In [[measure theory|measure-theoretic]] terms, we use the random variable <math>X</math> to "push-forward" the measure <math>P</math> on <math>\Omega</math> to a measure <math>p_X</math> on <math>\mathbb{R}</math>.
The underlying probability space <math>\Omega</math> is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as [[correlation and dependence]] or [[Independence (probability theory)|independence]] based on a [[joint distribution]] of two or more random variables on the same probability space. In practice, one often disposes of the space <math>\Omega</math> altogether and just puts a measure on <math>\mathbb{R}</math> that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on [[quantile function]]s for fuller development.
==Examples==
===Discrete random variable===
In an experiment a person may be chosen at random, and one random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to the person's height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.
Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum <math>\operatorname{PMF}(0) + \operatorname{PMF}(2) + \operatorname{PMF}(4) + \cdots</math>.
In examples such as these, the [[sample space]] is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.
If <math display = "inline">\{a_n\}, \{b_n\}</math> are countable sets of real numbers, <math display="inline">b_n >0</math> and <math>\sum_n b_n=1</math>, then <math> F=\sum_n b_n \delta_{a_n}</math> is a discrete distribution function. Here <math> \delta_t(x) = 0</math> for <math> x < t</math>, <math> \delta_t(x) = 1</math> for <math> x \ge t</math>. Taking for instance an enumeration of all rational numbers as <math>\{a_n\}</math>, one gets a discrete distribution function that is not a step function or piecewise constant.<ref name="Yates" />
====Coin toss====
The possible outcomes for one coin toss can be described by the sample space <math>\Omega = \{\text{heads}, \text{tails}\}</math>. We can introduce a real-valued random variable <math>Y</math> that models a $1 payoff for a successful bet on heads as follows:
:<math>
Y(\omega) =
\begin{cases}
1, & \text{if } \omega = \text{heads}, \\[6pt]
0, & \text{if } \omega = \text{tails}.
\end{cases}
</math>
If the coin is a [[fair coin]], ''Y'' has a [[probability mass function]] <math>f_Y</math> given by:
:<math>
f_Y(y) =
\begin{cases}
\tfrac 12,& \text{if }y=1,\\[6pt]
\tfrac 12,& \text{if }y=0,
\end{cases}
</math>
====Dice roll====
[[File:Dice Distribution (bar).svg| right | thumb | If the sample space is the set of possible numbers rolled on two dice, and the random variable of interest is the sum ''S'' of the numbers on the two dice, then ''S'' is a discrete random variable whose distribution is described by the [[probability mass function]] plotted as the height of picture columns here.]]
A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers ''n''<sub>1</sub> and ''n''<sub>2</sub> from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable ''X'' given by the function that maps the pair to the sum:
:<math>X((n_1, n_2)) = n_1 + n_2</math>
and (if the dice are [[fair die|fair]]) has a probability mass function ''ƒ''<sub>''X''</sub> given by:
:<math>f_X(S) = \frac{\min(S-1, 13-S)}{36}, \text{ for } S \in \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}</math>
===Continuous random variable===
Formally, a continuous random variable is a random variable whose [[cumulative distribution function]] is [[Continuous function|continuous]] everywhere.<ref name=":0">{{Cite book|title=Introduction to Probability|last=Bertsekas|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν.|isbn=188652940X|___location=Belmont, Mass.|oclc=51441829}}</ref> There are no "[[Discontinuity (mathematics)#Jump discontinuity|gaps]]", which would correspond to numbers which have a finite probability of [[Outcome (probability)|occurring]]. Instead, continuous random variables [[almost never]] take an exact prescribed value ''c'' (formally, <math display="inline">\forall c \in \mathbb{R}:\; \Pr(X = c) = 0</math>) but there is a positive probability that its value will lie in particular [[Interval (mathematics)|intervals]] which can be [[arbitrarily small]]. Continuous random variables usually admit [[probability density function]]s (PDF), which characterize their CDF and [[probability measure]]s;
such distributions are also called [[Absolutely continuous random variable|absolutely continuous]]; but some continuous distributions are [[Singular distribution|singular]], or mixes of an absolutely continuous part and a singular part.
An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, '''''X''''' = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any ''range'' of values. For example, the probability of choosing a number in [0, 180] is {{frac|1|2}}. Instead of speaking of a probability mass function, we say that the probability ''density'' of '''''X''''' is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.
More formally, given any [[Interval (mathematics)|interval]] <math display="inline">I = [a, b] = \{x \in \mathbb{R} : a \le x \le b \}</math>, a random variable <math>X_I \sim \operatorname{U}(I) = \operatorname{U}[a, b]</math> is called a "[[Continuous uniform distribution|continuous uniform]] random variable" (CURV) if the probability that it takes a value in a [[subinterval]] depends only on the length of the subinterval. This implies that the probability of <math>X_I</math> falling in any subinterval <math>[c, d] \sube [a, b]</math> is [[Proportionality (mathematics)|proportional]] to the [[Lebesgue measure|length]] of the subinterval, that is, if {{math|''a'' ≤ ''c'' ≤ ''d'' ≤ ''b''}}, one has
<math display="block">
\Pr\left( X_I \in [c,d]\right)
= \frac{d - c}{b - a}\Pr\left( X_I \in I\right)= \frac{d - c}{b - a}</math>
where the last equality results from the [[Probability axioms#Unitarity|unitarity axiom]] of probability. The [[probability density function]] of a CURV <math>X \sim \operatorname {U}[a, b]</math> is given by the [[indicator function]] of its interval of [[Support (mathematics)|support]] normalized by the interval's length: <math display="block">f_X(x) = \begin{cases}
\displaystyle{1 \over b-a}, & a \le x \le b \\
0, & \text{otherwise}.
\end{cases}</math>Of particular interest is the uniform distribution on the [[unit interval]] <math>[0, 1]</math>. Samples of any desired [[probability distribution]] <math>\operatorname{D}</math> can be generated by calculating the [[quantile function]] of <math>\operatorname{D}</math> on a [[Random number generation|randomly-generated number]] distributed uniformly on the unit interval. This exploits [[Cumulative distribution function#Properties|properties of cumulative distribution functions]], which are a unifying framework for all random variables.
===Mixed type===
A '''mixed random variable''' is a random variable whose [[cumulative distribution function]] is neither [[Piecewise constant|piecewise-constant]] (a discrete random variable) nor [[Continuous function|everywhere-continuous]].<ref name=":0" /> It can be realized as the sum of a discrete random variable and a continuous random variable; in which case the {{Abbr|CDF|cumulative distribution function}} will be the weighted average of the CDFs of the component variables.<ref name=":0" />
An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, '''''X''''' = −1; otherwise '''''X''''' = the value of the spinner as in the preceding example. There is a probability of {{frac|1|2}} that this random variable will have the value −1. Other ranges of values would have half the probabilities of the last example.
Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see {{Section link|Lebesgue's decomposition theorem|Refinement}}. The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
==Measure-theoretic definition==
The most formal, [[axiomatic]] definition of a random variable involves [[measure theory]]. Continuous random variables are defined in terms of [[set (mathematics)|set]]s of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the [[Banach–Tarski paradox]]) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a [[sigma-algebra]] to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the [[Borel σ-algebra]], which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or [[countably infinite]] number of [[union (set theory)|union]]s and/or [[intersection (set theory)|intersection]]s of such intervals.<ref name="UCSB" />
The measure-theoretic definition is as follows.
Let <math>(\Omega, \mathcal{F}, P)</math> be a [[probability space]] and <math>(E, \mathcal{E})</math> a [[measurable space]]. Then an '''<math>(E, \mathcal{E})</math>-valued random variable''' is a measurable function <math>X\colon \Omega \to E</math>, which means that, for every subset <math>B\in\mathcal{E}</math>, its [[preimage]] <math>X^{-1}(B)\in \mathcal{F}</math> where <math>X^{-1}(B) = \{\omega : X(\omega)\in B\}</math>.<ref>{{harvtxt|Fristedt|Gray|1996|loc=page 11}}</ref> This definition enables us to measure any subset <math>B\in \mathcal{E}</math> in the target space by looking at its preimage, which by assumption is measurable.
In more intuitive terms, a member of <math>\Omega</math> is a possible outcome, a member of <math>\mathcal{F}</math> is a measurable subset of possible outcomes, the function <math>P</math> gives the probability of each such measurable subset, <math>E</math> represents the set of values that the random variable can take (such as the set of real numbers), and a member of <math>\mathcal{E}</math> is a "well-behaved" (measurable) subset of <math>E</math> (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.
When <math>E</math> is a [[topological space]], then the most common choice for the [[σ-algebra]] <math>\mathcal{E}</math> is the [[Borel σ-algebra]] <math>\mathcal{B}(E)</math>, which is the σ-algebra generated by the collection of all open sets in <math>E</math>. In such case the <math>(E, \mathcal{E})</math>-valued random variable is called an '''<math>E</math>-valued random variable'''. Moreover, when the space <math>E</math> is the real line <math>\mathbb{R}</math>, then such a real-valued random variable is called simply a '''random variable'''.
===Real-valued random variables===
In this case the observation space is the set of real numbers. Recall, <math>(\Omega, \mathcal{F}, P)</math> is the probability space. For a real observation space, the function <math>X\colon \Omega \rightarrow \mathbb{R}</math> is a real-valued random variable if
:<math>\{ \omega : X(\omega) \le r \} \in \mathcal{F} \qquad \forall r \in \mathbb{R}.</math>
This definition is a special case of the above because the set <math>\{(-\infty, r]: r \in \R\}</math> generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that <math>\{ \omega : X(\omega) \le r \} = X^{-1}((-\infty, r])</math>.
==Moments==
The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of [[expected value]] of a random variable, denoted <math>\operatorname{E}[X]</math>, and also called the '''first [[Moment (mathematics)|moment]].''' In general, <math>\operatorname{E}[f(X)]</math> is not equal to <math>f(\operatorname{E}[X])</math>. Once the "average value" is known, one could then ask how far from this average value the values of <math>X</math> typically are, a question that is answered by the [[variance]] and [[standard deviation]] of a random variable. <math>\operatorname{E}[X]</math> can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of <math>X</math>.
Mathematically, this is known as the (generalised) [[problem of moments]]: for a given class of random variables <math>X</math>, find a collection <math>\{f_i\}</math> of functions such that the expectation values <math>\operatorname{E}[f_i(X)]</math> fully characterise the [[Probability distribution|distribution]] of the random variable <math>X</math>.
Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function <math>f(X)=X</math> of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a [[categorical variable|categorical]] random variable ''X'' that can take on the [[nominal data|nominal]] values "red", "blue" or "green", the real-valued function <math>[X = \text{green}]</math> can be constructed; this uses the [[Iverson bracket]], and has the value 1 if <math>X</math> has the value "green", 0 otherwise. Then, the [[expected value]] and other moments of this function can be determined.
==Functions of random variables==
A new random variable ''Y'' can be defined by [[Function composition|applying]] a real [[Measurable function|Borel measurable function]] <math>g\colon \mathbb{R} \rightarrow \mathbb{R}</math> to the outcomes of a [[real-valued]] random variable <math>X</math>. That is, <math>Y=g(X)</math>. The [[cumulative distribution function]] of <math>Y</math> is then
:<math>F_Y(y) = \operatorname{P}(g(X) \le y).</math>
If function <math>g</math> is invertible (i.e., <math>h = g^{-1}</math> exists, where <math>h</math> is <math>g</math>'s [[inverse function]]) and is either [[Monotonic function|increasing or decreasing]], then the previous relation can be extended to obtain
:<math>F_Y(y) = \operatorname{P}(g(X) \le y) =
\begin{cases}
\operatorname{P}(X \le h(y)) = F_X(h(y)),
& \text{if } h = g^{-1} \text{ increasing} ,\\
\\
\operatorname{P}(X \ge h(y)) = 1 - F_X(h(y)),
& \text{if } h = g^{-1} \text{ decreasing} .
\end{cases}</math>
With the same hypotheses of invertibility of <math>g</math>, assuming also [[differentiability]], the relation between the [[probability density function]]s can be found by differentiating both sides of the above expression with respect to <math>y</math>, in order to obtain<ref name=":0" />
:<math>f_Y(y) = f_X\bigl(h(y)\bigr) \left| \frac{d h(y)}{d y} \right|.</math>
If there is no invertibility of <math>g</math> but each <math>y</math> admits at most a countable number of roots (i.e., a finite, or countably infinite, number of <math>x_i</math> such that <math>y = g(x_i)</math>) then the previous relation between the [[probability density function]]s can be generalized with
:<math>f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right| </math>
where <math>x_i = g_i^{-1}(y)</math>, according to the [[inverse function theorem]]. The formulas for densities do not demand <math>g</math> to be increasing.
In the measure-theoretic, [[Probability axioms|axiomatic approach]] to probability, if a random variable <math>X</math> on <math>\Omega</math> and a [[measurable function|Borel measurable function]] <math>g\colon \mathbb{R} \rightarrow \mathbb{R}</math>, then <math>Y = g(X)</math> is also a random variable on <math>\Omega</math>, since the composition of measurable functions [[Closure (mathematics)|is also measurable]]. (However, this is not necessarily true if <math>g</math> is [[Lebesgue measurable]].{{Citation needed|date=October 2018}}) The same procedure that allowed one to go from a probability space <math>(\Omega, P) </math> to <math>(\mathbb{R}, dF_{X})</math> can be used to obtain the distribution of <math>Y</math>.
===Example 1===
Let <math>X</math> be a real-valued, [[continuous random variable]] and let <math>Y = X^2</math>.
:<math>F_Y(y) = \operatorname{P}(X^2 \le y).</math>
If <math>y < 0</math>, then <math>P(X^2 \leq y) = 0</math>, so
:<math>F_Y(y) = 0\qquad\hbox{if}\quad y < 0.</math>
If <math>y \geq 0</math>, then
:<math>\operatorname{P}(X^2 \le y) = \operatorname{P}(|X| \le \sqrt{y})
= \operatorname{P}(-\sqrt{y} \le X \le \sqrt{y}),</math>
so
:<math>F_Y(y) = F_X(\sqrt{y}) - F_X(-\sqrt{y})\qquad\hbox{if}\quad y \ge 0.</math>
===Example 2===
Suppose <math>X</math> is a random variable with a cumulative distribution
:<math> F_{X}(x) = P(X \leq x) = \frac{1}{(1 + e^{-x})^{\theta}}</math>
where <math>\theta > 0</math> is a fixed parameter. Consider the random variable <math> Y = \mathrm{log}(1 + e^{-X}).</math> Then,
:<math> F_{Y}(y) = P(Y \leq y) = P(\mathrm{log}(1 + e^{-X}) \leq y) = P(X \geq -\mathrm{log}(e^{y} - 1)).\,</math>
The last expression can be calculated in terms of the cumulative distribution of <math>X,</math> so
:<math>
\begin{align}
F_Y(y) & = 1 - F_X(-\log(e^y - 1)) \\[5pt]
& = 1 - \frac{1}{(1 + e^{\log(e^y - 1)})^\theta} \\[5pt]
& = 1 - \frac{1}{(1 + e^y - 1)^\theta} \\[5pt]
& = 1 - e^{-y \theta}.
\end{align}
</math>
which is the [[cumulative distribution function]] (CDF) of an [[exponential distribution]].
===Example 3===
Suppose <math>X</math> is a random variable with a [[standard normal distribution]], whose density is
:<math> f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}.</math>
Consider the random variable <math> Y = X^2.</math> We can find the density using the above formula for a change of variables:
:<math>f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right|. </math>
In this case the change is not [[Monotonic function|monotonic]], because every value of <math>Y</math> has two corresponding values of <math>X</math> (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,
:<math>f_Y(y) = 2f_X(g^{-1}(y)) \left| \frac{d g^{-1}(y)}{d y} \right|.</math>
The inverse transformation is
:<math>x = g^{-1}(y) = \sqrt{y}</math>
and its derivative is
:<math>\frac{d g^{-1}(y)}{d y} = \frac{1}{2\sqrt{y}} .</math>
Then,
:<math> f_Y(y) = 2\frac{1}{\sqrt{2\pi}}e^{-y/2} \frac{1}{2\sqrt{y}} = \frac{1}{\sqrt{2\pi y}}e^{-y/2}. </math>
This is a [[chi-squared distribution]] with one [[Degrees of freedom (statistics)|degree of freedom]].
===Example 4===
Suppose <math>X</math> is a random variable with a [[normal distribution]], whose density is
:<math> f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/(2\sigma^2)}.</math>
Consider the random variable <math> Y = X^2.</math> We can find the density using the above formula for a change of variables:
:<math>f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right|. </math>
In this case the change is not [[monotonic]], because every value of <math>Y</math> has two corresponding values of <math>X</math> (one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:
:<math>f_Y(y) = f_X(g_1^{-1}(y))\left|\frac{d g_1^{-1}(y)}{d y} \right| +f_X(g_2^{-1}(y))\left| \frac{d g_2^{-1}(y)}{d y} \right|.</math>
The inverse transformation is
:<math>x = g_{1,2}^{-1}(y) = \pm \sqrt{y}</math>
and its derivative is
:<math>\frac{d g_{1,2}^{-1}(y)}{d y} = \pm \frac{1}{2\sqrt{y}} .</math>
Then,
:<math> f_Y(y) = \frac{1}{\sqrt{2\pi\sigma^2}} \frac{1}{2\sqrt{y}} (e^{-(\sqrt{y}-\mu)^2/(2\sigma^2)}+e^{-(-\sqrt{y}-\mu)^2/(2\sigma^2)}) . </math>
This is a [[noncentral chi-squared distribution]] with one [[degree of freedom (statistics)|degree of freedom]].
==Some properties==
* The probability distribution of the sum of two independent random variables is the '''[[convolution]]''' of each of their distributions.
* Probability distributions are not a [[vector space]]—they are not closed under [[linear combination]]s, as these do not preserve non-negativity or total integral 1—but they are closed under [[convex combination]], thus forming a [[convex subset]] of the space of functions (or measures).
==Equivalence of random variables==
There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.
In increasing order of strength, the precise definition of these notions of equivalence is given below.
===Equality in distribution===
If the sample space is a subset of the real line, random variables ''X'' and ''Y'' are ''equal in distribution'' (denoted <math>X \stackrel{d}{=} Y</math>) if they have the same distribution functions:
:<math>\operatorname{P}(X \le x) = \operatorname{P}(Y \le x)\quad\text{for all }x.</math>
To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal [[moment generating function]]s have the same distribution. This provides, for example, a useful method of checking equality of certain functions of [[Independent and identically distributed random variables|independent, identically distributed (IID) random variables]]. However, the moment generating function exists only for distributions that have a defined [[Laplace transform]].
===Almost sure equality===
Two random variables ''X'' and ''Y'' are ''equal [[almost surely]]'' (denoted <math>X \; \stackrel{\text{a.s.}}{=} \; Y</math>) if, and only if, the probability that they are different is [[Null set|zero]]:
:<math>\operatorname{P}(X \neq Y) = 0.</math>
For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
:<math>d_\infty(X,Y)=\operatorname{ess} \sup_\omega|X(\omega)-Y(\omega)|,</math>
where "ess sup" represents the [[essential supremum]] in the sense of [[measure theory]].
===Equality===
Finally, the two random variables ''X'' and ''Y'' are ''equal'' if they are equal as functions on their measurable space:
:<math>X(\omega)=Y(\omega)\qquad\hbox{for all }\omega.</math>
This notion is typically the least useful in probability theory because in practice and in theory, the underlying [[measure space]] of the [[Experiment (probability theory)|experiment]] is rarely explicitly characterized or even characterizable.
==Convergence==
{{Main|Convergence of random variables}}
A significant theme in mathematical statistics consists of obtaining convergence results for certain [[sequence]]s of random variables; for instance the [[law of large numbers]] and the [[central limit theorem]].
There are various senses in which a sequence <math>X_n</math> of random variables can converge to a random variable <math>X</math>. These are explained in the article on [[convergence of random variables]].
==See also==
{{Portal|Mathematics}}
{{Div col|colwidth=22em}}
*[[Aleatoricism]]
*[[Algebra of random variables]]
*[[Event (probability theory)]]
*[[Multivariate random variable]]
*[[Pairwise independence|Pairwise independent random variables]]
*[[Observable variable]]
*[[Random element]]
*[[Random function]]
*[[Random measure]]
*[[Random number generator]] produces a random value
*[[Random vector]]
*[[Randomness]]
*[[Stochastic process]]
*[[Relationships among probability distributions]]{{Div col end}}
==References==
=== Inline citations ===
{{Reflist}}
===Literature===
{{Refbegin}}
* {{cite book | last1 = Fristedt | first1 = Bert | last2 = Gray | first2 = Lawrence | title = A modern approach to probability theory | year = 1996 | publisher = Birkhäuser | ___location = Boston | url = https://books.google.com/books/about/A_Modern_Approach_to_Probability_Theory.html?id=5D5O8xyM-kMC | isbn = 3-7643-3807-5 }}
* {{cite book | last = Kallenberg | first = Olav | author-link = Olav Kallenberg | year = 1986 | title = Random Measures | edition = 4th | publisher = [[Akademie Verlag]] | ___location = Berlin | mr = 0854102 | isbn = 0-12-394960-2 | url = https://books.google.com/books/about/Random_measures.html?id=bBnvAAAAMAAJ}}
* {{cite book | last = Kallenberg | first = Olav | year = 2001 | title = Foundations of Modern Probability | edition = 2nd | publisher = [[Springer Verlag]] | ___location = Berlin | isbn = 0-387-95313-2 | url = https://books.google.com/?id=L6fhXh13OyMC}}
* {{cite book | author-link = Athanasios Papoulis | last = Papoulis | first = Athanasios | year = 1965 | title = Probability, Random Variables, and Stochastic Processes | publisher = [[McGraw–Hill]] | ___location = Tokyo | edition = 9th | isbn = 0-07-119981-0 | url = http://www.mhhe.com/engcs/electrical/papoulis/}}
{{Refend}}
==External links==
*{{springer|title=Random variable|id=p/r077360}}
* {{citation | last = Zukerman| first = Moshe| year = 2014 | title = Introduction to Queueing Theory and Stochastic Teletraffic Models | url=http://www.ee.cityu.edu.hk/~zukerman/classnotes.pdf}}
* {{citation | last = Zukerman| first = Moshe| year = 2014| title = Basic Probability Topics | url=http://www.ee.cityu.edu.hk/~zukerman/probability.pdf}}
{{Statistics|state = collapsed}}
{{DEFAULTSORT:Random Variable}}
[[Category:Statistical randomness]]' |
New page wikitext, after the edit (new_wikitext) | '{{short description|Variable representing a random phenomenon}}
{{Probability fundamentals}}
In [[probability]] and [[statistics]], a '''random variable''', '''random quantity''', '''aleatory variable''', or '''stochastic variable''' is described informally as a [[Dependent and independent variables|variable whose values depend]] on [[Outcome (probability)|outcomes]] of a [[Randomness|random]] phenomenon.<ref>{{cite book|last1=Blitzstein|first1=Joe|last2=Hwang|first2=Jessica|title=Introduction to Probability|date=2014|publisher=CRC Press|isbn=9781466575592}}</ref> The formal mathematical treatment of random variables is a topic in [[probability theory]]. In that context, a random variable is understood as a [[measurable function]] defined on a [[probability space]] that maps from the [[sample space]] to the [[real number]]s.<ref name="UCSB">{{cite web | title = Economics 245A – Introduction to Measure Theory | url = http://econ.ucsb.edu/~doug/245a/Lectures/Measure%20Theory.pdf | last = Steigerwald | first = Douglas G. | publisher = University of California, Santa Barbara | access-date = April 26, 2013}}</ref>
[[File:Random Variable as a Function-en.svg|thumb|This graph shows how random variable is a function from all possible outcomes to real values. It also shows how random variable is used for defining probability mass functions.]]
A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, because of imprecise measurements or [[quantum uncertainty]]). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but is instead related to philosophical arguments over the [[interpretation of probability]]. The mathematics works the same regardless of the particular interpretation in use.
As a function, a random variable is required to be [[Measurable function|measurable]], which allows for probabilities to be assigned to sets of its potential values. It is common that the outcomes depend on some physical variables that are not predictable. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physical conditions, so the outcome being observed is uncertain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.
The [[___domain of a function|___domain]] of a random variable is called a ''sample space,'' defined as the set of possible outcomes of a non-deterministic event. For example, in the event of a coin toss, only two possible outcomes are possible: heads or tails.
A random variable has a [[probability distribution]], which specifies the probability of [[Borel subset]]s of its range. Random variables can be [[Discrete random variable|discrete]], that is, taking any of a specified finite or [[countable set|countable list]] of values (having a countable range), endowed with a [[probability mass function]] that is characteristic of the random variable's probability distribution; or [[Continuous random variable|continuous]], taking any numerical value in an interval or collection of intervals (having an [[Uncountable set|uncountable]] range), via a [[probability density function]] that is characteristic of the random variable's probability distribution; or a mixture of both.
Two random variables with the same probability distribution can still differ in terms of their associations with, or [[independence (probability theory)|independence]] from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called [[random variate]]s.
==Definition==
A '''random variable''' is a [[measurable function]] <math>X \colon \Omega \to E</math> from a set of possible [[outcome (probability)|outcome]]s <math> \Omega </math> to a [[measurable space]] <math> E</math>. The technical axiomatic definition requires <math>\Omega</math> to be a sample space of a [[probability space|probability triple]] <math>(\Omega, \mathcal{F}, \operatorname{P})</math> (see the [[#Measure-theoretic definition|measure-theoretic definition]]). A random variable is often denoted by capital [[Latin script|roman letters]] such as <math>X</math>, <math>Y</math>, <math>Z</math>, <math>T</math>.<ref name=":1">{{Cite web|date=2020-04-26|title=List of Probability and Statistics Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/probability-statistics-symbols/|access-date=2020-08-21|website=Math Vault|language=en-US}}</ref><ref>{{Cite web|title=Random Variables|url=https://www.mathsisfun.com/data/random-variables.html|access-date=2020-08-21|website=www.mathsisfun.com}}</ref>
The probability that <math>X</math> takes on a value in a measurable set <math>S\subseteq E</math> is written as
: <math>\operatorname{P}(X \in S) = \operatorname{P}(\{ \omega \in \Omega \mid X(\omega) \in S \})</math><ref name=":1" />
===Standard case===
In many cases, <math>X</math> is [[Real number|real-valued]], i.e. <math>E = \mathbb{R}</math>. In some contexts, the term [[random element]] (see [[#Extensions|extensions]]) is used to denote a random variable not of this form.
{{Anchor|Discrete random variable}}When the [[Image (mathematics)|image]] (or range) of <math>X</math> is [[countable set|countable]], the random variable is called a '''discrete random variable'''<ref name="Yates">{{cite book | last = Yates | first = Daniel S. | last2 = Moore | first2 = David S | last3 = Starnes | first3 = Daren S. | year = 2003 | title = The Practice of Statistics | edition = 2nd | publisher = [[W. H. Freeman and Company|Freeman]] | ___location = New York | url = http://bcs.whfreeman.com/yates2e/ | isbn = 978-0-7167-4773-4 | url-status = dead | archive-url = https://web.archive.org/web/20050209001108/http://bcs.whfreeman.com/yates2e/ | archive-date = 2005-02-09 }}</ref>{{rp|399}} and its distribution is a [[discrete probability distribution]], i.e. can be described by a [[probability mass function]] that assigns a probability to each value in the image of <math>X</math>. If the image is uncountably infinite (usually an [[Interval (mathematics)|interval]]) then <math>X</math> is called a '''continuous random variable'''.<ref>{{Cite web|title=Random Variables|url=http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm|access-date=2020-08-21|website=www.stat.yale.edu}}</ref>{{Citation needed|reason=Statistics 101 is hardly a sufficient citation, even coming from Yale|date=October 2020}} In the special case that it is [[absolutely continuous]], its distribution can be described by a [[probability density function]], which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous,<ref>{{cite book|author1=L. Castañeda |author2=V. Arunachalam |author3=S. Dharmaraja |name-list-style=amp |title = Introduction to Probability and Stochastic Processes with Applications | year = 2012 | publisher= Wiley | page = 67 | url=https://books.google.com/books?id=zxXRn-Qmtk8C&pg=PA67 |isbn=9781118344941 }}</ref> a [[mixture distribution]] is one such counterexample; such random variables cannot be described by a probability density or a probability mass function.
Any random variable can be described by its [[cumulative distribution function]], which describes the probability that the random variable will be less than or equal to a certain value.
===Extensions===
The term "random variable" in statistics is traditionally limited to the [[real number|real-valued]] case (<math>E=\mathbb{R}</math>). In this case, the structure of the real numbers makes it possible to define quantities such as the [[expected value]] and [[variance]] of a random variable, its [[cumulative distribution function]], and the [[moment (mathematics)|moment]]s of its distribution.
However, the definition above is valid for any [[measurable space]] <math>E</math> of values. Thus one can consider random elements of other sets <math>E</math>, such as random [[Boolean-valued function|boolean value]]s, [[categorical variable|categorical value]]s, [[Covariance matrix#Complex random vectors|complex numbers]], [[random vector|vector]]s, [[random matrix|matrices]], [[random sequence|sequence]]s, [[Tree (graph theory)|tree]]s, [[random compact set|set]]s, [[shape]]s, [[manifold]]s, and [[random function|function]]s. One may then specifically refer to a ''random variable of [[data type|type]] <math>E</math>'', or an ''<math>E</math>-valued random variable''.
This more general concept of a [[random element]] is particularly useful in disciplines such as [[graph theory]], [[machine learning]], [[natural language processing]], and other fields in [[discrete mathematics]] and [[computer science]], where one is often interested in modeling the random variation of non-numerical [[data structure]]s. In some cases, it is nonetheless convenient to represent each element of <math>E</math>, using one or more real numbers. In this case, a random element may optionally be represented as a [[random vector|vector of real-valued random variables]] (all defined on the same underlying probability space <math>\Omega</math>, which allows the different random variables to [[mutual information|covary]]). For example:
*A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are <math>(1 \ 0 \ 0 \ 0 \ \cdots)</math>, <math>(0 \ 1 \ 0 \ 0 \ \cdots)</math>, <math>(0 \ 0 \ 1 \ 0 \ \cdots)</math> and the position of the 1 indicates the word.
*A random sentence of given length <math>N</math> may be represented as a vector of <math>N</math> random words.
*A [[random graph]] on <math>N</math> given vertices may be represented as a <math>N \times N</math> matrix of random variables, whose values specify the [[adjacency matrix]] of the random graph.
*A [[random function]] <math>F</math> may be represented as a collection of random variables <math>F(x)</math>, giving the function's values at the various points <math>x</math> in the function's ___domain. The <math>F(x)</math> are ordinary real-valued random variables provided that the function is real-valued. For example, a [[stochastic process]] is a random function of time, a [[random vector]] is a random function of some index set such as <math>1,2,\ldots, n</math>, and [[random field]] is a random function on any set (typically time, space, or a discrete set).
==Distribution functions==
If a random variable <math>X\colon \Omega \to \mathbb{R}</math> defined on the probability space <math>(\Omega, \mathcal{F}, \operatorname{P})</math> is given, we can ask questions like "How likely is it that the value of <math>X</math> is equal to 2?". This is the same as the probability of the event <math>\{ \omega : X(\omega) = 2 \}\,\! </math> which is often written as <math>P(X = 2)\,\!</math> or <math>p_X(2)</math> for short.
Recording all these probabilities of output ranges of a real-valued random variable <math>X</math> yields the [[probability distribution]] of <math>X</math>. The probability distribution "forgets" about the particular probability space used to define <math>X</math> and only records the probabilities of various values of <math>X</math>. Such a probability distribution can always be captured by its [[cumulative distribution function]]
:<math>F_X(x) = \operatorname{P}(X \le x)</math>
and sometimes also using a [[probability density function]], <math>p_X</math>. In [[measure theory|measure-theoretic]] terms, we use the random variable <math>X</math> to "push-forward" the measure <math>P</math> on <math>\Omega</math> to a measure <math>p_X</math> on <math>\mathbb{R}</math>.
The underlying probability space <math>\Omega</math> is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as [[correlation and dependence]] or [[Independence (probability theory)|independence]] based on a [[joint distribution]] of two or more random variables on the same probability space. In practice, one often disposes of the space <math>\Omega</math> altogether and just puts a measure on <math>\mathbb{R}</math> that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on [[quantile function]]s for fuller development.
==Examples==
===Discrete random variable===
In an experiment a person may be chosen at random, and one random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to the person's height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.
Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum <math>\operatorname{PMF}(0) + \operatorname{PMF}(2) + \operatorname{PMF}(4) + \cdots</math>.
In examples such as these, the [[sample space]] is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.
If <math display = "inline">\{a_n\}, \{b_n\}</math> are countable sets of real numbers, <math display="inline">b_n >0</math> and <math>\sum_n b_n=1</math>, then <math> F=\sum_n b_n \delta_{a_n}</math> is a discrete distribution function. Here <math> \delta_t(x) = 0</math> for <math> x < t</math>, <math> \delta_t(x) = 1</math> for <math> x \ge t</math>. Taking for instance an enumeration of all rational numbers as <math>\{a_n\}</math>, one gets a discrete distribution function that is not a step function or piecewise constant.<ref name="Yates" />
====Coin toss====
The possible outcomes for one coin toss can be described by the sample space <math>\Omega = \{\text{heads}, \text{tails}\}</math>. We can introduce a real-valued random variable <math>Y</math> that models a $1 payoff for a successful bet on heads as follows:
:<math>
Y(\omega) =
\begin{cases}
1, & \text{if } \omega = \text{heads}, \\[6pt]
0, & \text{if } \omega = \text{tails}.
\end{cases}
</math>
If the coin is a [[fair coin]], ''Y'' has a [[probability mass function]] <math>f_Y</math> given by:
:<math>
f_Y(y) =
\begin{cases}
\tfrac 12,& \text{if }y=1,\\[6pt]
\tfrac 12,& \text{if }y=0,
\end{cases}
</math>
====Dice roll====
dodo
A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers ''n''<sub>1</sub> and ''n''<sub>2</sub> from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable ''X'' given by the function that maps the pair to the sum:
:<math>X((n_1, n_2)) = n_1 + n_2</math>
and (if the dice are [[fair die|fair]]) has a probability mass function ''ƒ''<sub>''X''</sub> given by:
:<math>f_X(S) = \frac{\min(S-1, 13-S)}{36}, \text{ for } S \in \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}</math>
===Continuous random variable===
Formally, a continuous random variable is a random variable whose [[cumulative distribution function]] is [[Continuous function|continuous]] everywhere.<ref name=":0">{{Cite book|title=Introduction to Probability|last=Bertsekas|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν.|isbn=188652940X|___location=Belmont, Mass.|oclc=51441829}}</ref> There are no "[[Discontinuity (mathematics)#Jump discontinuity|gaps]]", which would correspond to numbers which have a finite probability of [[Outcome (probability)|occurring]]. Instead, continuous random variables [[almost never]] take an exact prescribed value ''c'' (formally, <math display="inline">\forall c \in \mathbb{R}:\; \Pr(X = c) = 0</math>) but there is a positive probability that its value will lie in particular [[Interval (mathematics)|intervals]] which can be [[arbitrarily small]]. Continuous random variables usually admit [[probability density function]]s (PDF), which characterize their CDF and [[probability measure]]s;
such distributions are also called [[Absolutely continuous random variable|absolutely continuous]]; but some continuous distributions are [[Singular distribution|singular]], or mixes of an absolutely continuous part and a singular part.
An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, '''''X''''' = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any ''range'' of values. For example, the probability of choosing a number in [0, 180] is {{frac|1|2}}. Instead of speaking of a probability mass function, we say that the probability ''density'' of '''''X''''' is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.
More formally, given any [[Interval (mathematics)|interval]] <math display="inline">I = [a, b] = \{x \in \mathbb{R} : a \le x \le b \}</math>, a random variable <math>X_I \sim \operatorname{U}(I) = \operatorname{U}[a, b]</math> is called a "[[Continuous uniform distribution|continuous uniform]] random variable" (CURV) if the probability that it takes a value in a [[subinterval]] depends only on the length of the subinterval. This implies that the probability of <math>X_I</math> falling in any subinterval <math>[c, d] \sube [a, b]</math> is [[Proportionality (mathematics)|proportional]] to the [[Lebesgue measure|length]] of the subinterval, that is, if {{math|''a'' ≤ ''c'' ≤ ''d'' ≤ ''b''}}, one has
<math display="block">
\Pr\left( X_I \in [c,d]\right)
= \frac{d - c}{b - a}\Pr\left( X_I \in I\right)= \frac{d - c}{b - a}</math>
where the last equality results from the [[Probability axioms#Unitarity|unitarity axiom]] of probability. The [[probability density function]] of a CURV <math>X \sim \operatorname {U}[a, b]</math> is given by the [[indicator function]] of its interval of [[Support (mathematics)|support]] normalized by the interval's length: <math display="block">f_X(x) = \begin{cases}
\displaystyle{1 \over b-a}, & a \le x \le b \\
0, & \text{otherwise}.
\end{cases}</math>Of particular interest is the uniform distribution on the [[unit interval]] <math>[0, 1]</math>. Samples of any desired [[probability distribution]] <math>\operatorname{D}</math> can be generated by calculating the [[quantile function]] of <math>\operatorname{D}</math> on a [[Random number generation|randomly-generated number]] distributed uniformly on the unit interval. This exploits [[Cumulative distribution function#Properties|properties of cumulative distribution functions]], which are a unifying framework for all random variables.
===Mixed type===
A '''mixed random variable''' is a random variable whose [[cumulative distribution function]] is neither [[Piecewise constant|piecewise-constant]] (a discrete random variable) nor [[Continuous function|everywhere-continuous]].<ref name=":0" /> It can be realized as the sum of a discrete random variable and a continuous random variable; in which case the {{Abbr|CDF|cumulative distribution function}} will be the weighted average of the CDFs of the component variables.<ref name=":0" />
An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, '''''X''''' = −1; otherwise '''''X''''' = the value of the spinner as in the preceding example. There is a probability of {{frac|1|2}} that this random variable will have the value −1. Other ranges of values would have half the probabilities of the last example.
Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see {{Section link|Lebesgue's decomposition theorem|Refinement}}. The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
==Measure-theoretic definition==
The most formal, [[axiomatic]] definition of a random variable involves [[measure theory]]. Continuous random variables are defined in terms of [[set (mathematics)|set]]s of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the [[Banach–Tarski paradox]]) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a [[sigma-algebra]] to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the [[Borel σ-algebra]], which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or [[countably infinite]] number of [[union (set theory)|union]]s and/or [[intersection (set theory)|intersection]]s of such intervals.<ref name="UCSB" />
The measure-theoretic definition is as follows.
Let <math>(\Omega, \mathcal{F}, P)</math> be a [[probability space]] and <math>(E, \mathcal{E})</math> a [[measurable space]]. Then an '''<math>(E, \mathcal{E})</math>-valued random variable''' is a measurable function <math>X\colon \Omega \to E</math>, which means that, for every subset <math>B\in\mathcal{E}</math>, its [[preimage]] <math>X^{-1}(B)\in \mathcal{F}</math> where <math>X^{-1}(B) = \{\omega : X(\omega)\in B\}</math>.<ref>{{harvtxt|Fristedt|Gray|1996|loc=page 11}}</ref> This definition enables us to measure any subset <math>B\in \mathcal{E}</math> in the target space by looking at its preimage, which by assumption is measurable.
In more intuitive terms, a member of <math>\Omega</math> is a possible outcome, a member of <math>\mathcal{F}</math> is a measurable subset of possible outcomes, the function <math>P</math> gives the probability of each such measurable subset, <math>E</math> represents the set of values that the random variable can take (such as the set of real numbers), and a member of <math>\mathcal{E}</math> is a "well-behaved" (measurable) subset of <math>E</math> (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.
When <math>E</math> is a [[topological space]], then the most common choice for the [[σ-algebra]] <math>\mathcal{E}</math> is the [[Borel σ-algebra]] <math>\mathcal{B}(E)</math>, which is the σ-algebra generated by the collection of all open sets in <math>E</math>. In such case the <math>(E, \mathcal{E})</math>-valued random variable is called an '''<math>E</math>-valued random variable'''. Moreover, when the space <math>E</math> is the real line <math>\mathbb{R}</math>, then such a real-valued random variable is called simply a '''random variable'''.
===Real-valued random variables===
In this case the observation space is the set of real numbers. Recall, <math>(\Omega, \mathcal{F}, P)</math> is the probability space. For a real observation space, the function <math>X\colon \Omega \rightarrow \mathbb{R}</math> is a real-valued random variable if
:<math>\{ \omega : X(\omega) \le r \} \in \mathcal{F} \qquad \forall r \in \mathbb{R}.</math>
This definition is a special case of the above because the set <math>\{(-\infty, r]: r \in \R\}</math> generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that <math>\{ \omega : X(\omega) \le r \} = X^{-1}((-\infty, r])</math>.
==Moments==
The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of [[expected value]] of a random variable, denoted <math>\operatorname{E}[X]</math>, and also called the '''first [[Moment (mathematics)|moment]].''' In general, <math>\operatorname{E}[f(X)]</math> is not equal to <math>f(\operatorname{E}[X])</math>. Once the "average value" is known, one could then ask how far from this average value the values of <math>X</math> typically are, a question that is answered by the [[variance]] and [[standard deviation]] of a random variable. <math>\operatorname{E}[X]</math> can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of <math>X</math>.
Mathematically, this is known as the (generalised) [[problem of moments]]: for a given class of random variables <math>X</math>, find a collection <math>\{f_i\}</math> of functions such that the expectation values <math>\operatorname{E}[f_i(X)]</math> fully characterise the [[Probability distribution|distribution]] of the random variable <math>X</math>.
Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function <math>f(X)=X</math> of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a [[categorical variable|categorical]] random variable ''X'' that can take on the [[nominal data|nominal]] values "red", "blue" or "green", the real-valued function <math>[X = \text{green}]</math> can be constructed; this uses the [[Iverson bracket]], and has the value 1 if <math>X</math> has the value "green", 0 otherwise. Then, the [[expected value]] and other moments of this function can be determined.
==Functions of random variables==
A new random variable ''Y'' can be defined by [[Function composition|applying]] a real [[Measurable function|Borel measurable function]] <math>g\colon \mathbb{R} \rightarrow \mathbb{R}</math> to the outcomes of a [[real-valued]] random variable <math>X</math>. That is, <math>Y=g(X)</math>. The [[cumulative distribution function]] of <math>Y</math> is then
:<math>F_Y(y) = \operatorname{P}(g(X) \le y).</math>
If function <math>g</math> is invertible (i.e., <math>h = g^{-1}</math> exists, where <math>h</math> is <math>g</math>'s [[inverse function]]) and is either [[Monotonic function|increasing or decreasing]], then the previous relation can be extended to obtain
:<math>F_Y(y) = \operatorname{P}(g(X) \le y) =
\begin{cases}
\operatorname{P}(X \le h(y)) = F_X(h(y)),
& \text{if } h = g^{-1} \text{ increasing} ,\\
\\
\operatorname{P}(X \ge h(y)) = 1 - F_X(h(y)),
& \text{if } h = g^{-1} \text{ decreasing} .
\end{cases}</math>
With the same hypotheses of invertibility of <math>g</math>, assuming also [[differentiability]], the relation between the [[probability density function]]s can be found by differentiating both sides of the above expression with respect to <math>y</math>, in order to obtain<ref name=":0" />
:<math>f_Y(y) = f_X\bigl(h(y)\bigr) \left| \frac{d h(y)}{d y} \right|.</math>
If there is no invertibility of <math>g</math> but each <math>y</math> admits at most a countable number of roots (i.e., a finite, or countably infinite, number of <math>x_i</math> such that <math>y = g(x_i)</math>) then the previous relation between the [[probability density function]]s can be generalized with
:<math>f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right| </math>
where <math>x_i = g_i^{-1}(y)</math>, according to the [[inverse function theorem]]. The formulas for densities do not demand <math>g</math> to be increasing.
In the measure-theoretic, [[Probability axioms|axiomatic approach]] to probability, if a random variable <math>X</math> on <math>\Omega</math> and a [[measurable function|Borel measurable function]] <math>g\colon \mathbb{R} \rightarrow \mathbb{R}</math>, then <math>Y = g(X)</math> is also a random variable on <math>\Omega</math>, since the composition of measurable functions [[Closure (mathematics)|is also measurable]]. (However, this is not necessarily true if <math>g</math> is [[Lebesgue measurable]].{{Citation needed|date=October 2018}}) The same procedure that allowed one to go from a probability space <math>(\Omega, P) </math> to <math>(\mathbb{R}, dF_{X})</math> can be used to obtain the distribution of <math>Y</math>.
===Example 1===
Let <math>X</math> be a real-valued, [[continuous random variable]] and let <math>Y = X^2</math>.
:<math>F_Y(y) = \operatorname{P}(X^2 \le y).</math>
If <math>y < 0</math>, then <math>P(X^2 \leq y) = 0</math>, so
:<math>F_Y(y) = 0\qquad\hbox{if}\quad y < 0.</math>
If <math>y \geq 0</math>, then
:<math>\operatorname{P}(X^2 \le y) = \operatorname{P}(|X| \le \sqrt{y})
= \operatorname{P}(-\sqrt{y} \le X \le \sqrt{y}),</math>
so
:<math>F_Y(y) = F_X(\sqrt{y}) - F_X(-\sqrt{y})\qquad\hbox{if}\quad y \ge 0.</math>
===Example 2===
Suppose <math>X</math> is a random variable with a cumulative distribution
:<math> F_{X}(x) = P(X \leq x) = \frac{1}{(1 + e^{-x})^{\theta}}</math>
where <math>\theta > 0</math> is a fixed parameter. Consider the random variable <math> Y = \mathrm{log}(1 + e^{-X}).</math> Then,
:<math> F_{Y}(y) = P(Y \leq y) = P(\mathrm{log}(1 + e^{-X}) \leq y) = P(X \geq -\mathrm{log}(e^{y} - 1)).\,</math>
The last expression can be calculated in terms of the cumulative distribution of <math>X,</math> so
:<math>
\begin{align}
F_Y(y) & = 1 - F_X(-\log(e^y - 1)) \\[5pt]
& = 1 - \frac{1}{(1 + e^{\log(e^y - 1)})^\theta} \\[5pt]
& = 1 - \frac{1}{(1 + e^y - 1)^\theta} \\[5pt]
& = 1 - e^{-y \theta}.
\end{align}
</math>
which is the [[cumulative distribution function]] (CDF) of an [[exponential distribution]].
===Example 3===
Suppose <math>X</math> is a random variable with a [[standard normal distribution]], whose density is
:<math> f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}.</math>
Consider the random variable <math> Y = X^2.</math> We can find the density using the above formula for a change of variables:
:<math>f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right|. </math>
In this case the change is not [[Monotonic function|monotonic]], because every value of <math>Y</math> has two corresponding values of <math>X</math> (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,
:<math>f_Y(y) = 2f_X(g^{-1}(y)) \left| \frac{d g^{-1}(y)}{d y} \right|.</math>
The inverse transformation is
:<math>x = g^{-1}(y) = \sqrt{y}</math>
and its derivative is
:<math>\frac{d g^{-1}(y)}{d y} = \frac{1}{2\sqrt{y}} .</math>
Then,
:<math> f_Y(y) = 2\frac{1}{\sqrt{2\pi}}e^{-y/2} \frac{1}{2\sqrt{y}} = \frac{1}{\sqrt{2\pi y}}e^{-y/2}. </math>
This is a [[chi-squared distribution]] with one [[Degrees of freedom (statistics)|degree of freedom]].
===Example 4===
Suppose <math>X</math> is a random variable with a [[normal distribution]], whose density is
:<math> f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/(2\sigma^2)}.</math>
Consider the random variable <math> Y = X^2.</math> We can find the density using the above formula for a change of variables:
:<math>f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right|. </math>
In this case the change is not [[monotonic]], because every value of <math>Y</math> has two corresponding values of <math>X</math> (one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:
:<math>f_Y(y) = f_X(g_1^{-1}(y))\left|\frac{d g_1^{-1}(y)}{d y} \right| +f_X(g_2^{-1}(y))\left| \frac{d g_2^{-1}(y)}{d y} \right|.</math>
The inverse transformation is
:<math>x = g_{1,2}^{-1}(y) = \pm \sqrt{y}</math>
and its derivative is
:<math>\frac{d g_{1,2}^{-1}(y)}{d y} = \pm \frac{1}{2\sqrt{y}} .</math>
Then,
:<math> f_Y(y) = \frac{1}{\sqrt{2\pi\sigma^2}} \frac{1}{2\sqrt{y}} (e^{-(\sqrt{y}-\mu)^2/(2\sigma^2)}+e^{-(-\sqrt{y}-\mu)^2/(2\sigma^2)}) . </math>
This is a [[noncentral chi-squared distribution]] with one [[degree of freedom (statistics)|degree of freedom]].
==Some properties==
* The probability distribution of the sum of two independent random variables is the '''[[convolution]]''' of each of their distributions.
* Probability distributions are not a [[vector space]]—they are not closed under [[linear combination]]s, as these do not preserve non-negativity or total integral 1—but they are closed under [[convex combination]], thus forming a [[convex subset]] of the space of functions (or measures).
==Equivalence of random variables==
There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.
In increasing order of strength, the precise definition of these notions of equivalence is given below.
===Equality in distribution===
If the sample space is a subset of the real line, random variables ''X'' and ''Y'' are ''equal in distribution'' (denoted <math>X \stackrel{d}{=} Y</math>) if they have the same distribution functions:
:<math>\operatorname{P}(X \le x) = \operatorname{P}(Y \le x)\quad\text{for all }x.</math>
To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal [[moment generating function]]s have the same distribution. This provides, for example, a useful method of checking equality of certain functions of [[Independent and identically distributed random variables|independent, identically distributed (IID) random variables]]. However, the moment generating function exists only for distributions that have a defined [[Laplace transform]].
===Almost sure equality===
Two random variables ''X'' and ''Y'' are ''equal [[almost surely]]'' (denoted <math>X \; \stackrel{\text{a.s.}}{=} \; Y</math>) if, and only if, the probability that they are different is [[Null set|zero]]:
:<math>\operatorname{P}(X \neq Y) = 0.</math>
For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
:<math>d_\infty(X,Y)=\operatorname{ess} \sup_\omega|X(\omega)-Y(\omega)|,</math>
where "ess sup" represents the [[essential supremum]] in the sense of [[measure theory]].
===Equality===
Finally, the two random variables ''X'' and ''Y'' are ''equal'' if they are equal as functions on their measurable space:
:<math>X(\omega)=Y(\omega)\qquad\hbox{for all }\omega.</math>
This notion is typically the least useful in probability theory because in practice and in theory, the underlying [[measure space]] of the [[Experiment (probability theory)|experiment]] is rarely explicitly characterized or even characterizable.
==Convergence==
{{Main|Convergence of random variables}}
A significant theme in mathematical statistics consists of obtaining convergence results for certain [[sequence]]s of random variables; for instance the [[law of large numbers]] and the [[central limit theorem]].
There are various senses in which a sequence <math>X_n</math> of random variables can converge to a random variable <math>X</math>. These are explained in the article on [[convergence of random variables]].
==See also==
{{Portal|Mathematics}}
{{Div col|colwidth=22em}}
*[[Aleatoricism]]
*[[Algebra of random variables]]
*[[Event (probability theory)]]
*[[Multivariate random variable]]
*[[Pairwise independence|Pairwise independent random variables]]
*[[Observable variable]]
*[[Random element]]
*[[Random function]]
*[[Random measure]]
*[[Random number generator]] produces a random value
*[[Random vector]]
*[[Randomness]]
*[[Stochastic process]]
*[[Relationships among probability distributions]]{{Div col end}}
==References==
=== Inline citations ===
{{Reflist}}
===Literature===
{{Refbegin}}
* {{cite book | last1 = Fristedt | first1 = Bert | last2 = Gray | first2 = Lawrence | title = A modern approach to probability theory | year = 1996 | publisher = Birkhäuser | ___location = Boston | url = https://books.google.com/books/about/A_Modern_Approach_to_Probability_Theory.html?id=5D5O8xyM-kMC | isbn = 3-7643-3807-5 }}
* {{cite book | last = Kallenberg | first = Olav | author-link = Olav Kallenberg | year = 1986 | title = Random Measures | edition = 4th | publisher = [[Akademie Verlag]] | ___location = Berlin | mr = 0854102 | isbn = 0-12-394960-2 | url = https://books.google.com/books/about/Random_measures.html?id=bBnvAAAAMAAJ}}
* {{cite book | last = Kallenberg | first = Olav | year = 2001 | title = Foundations of Modern Probability | edition = 2nd | publisher = [[Springer Verlag]] | ___location = Berlin | isbn = 0-387-95313-2 | url = https://books.google.com/?id=L6fhXh13OyMC}}
* {{cite book | author-link = Athanasios Papoulis | last = Papoulis | first = Athanasios | year = 1965 | title = Probability, Random Variables, and Stochastic Processes | publisher = [[McGraw–Hill]] | ___location = Tokyo | edition = 9th | isbn = 0-07-119981-0 | url = http://www.mhhe.com/engcs/electrical/papoulis/}}
{{Refend}}
==External links==
*{{springer|title=Random variable|id=p/r077360}}
* {{citation | last = Zukerman| first = Moshe| year = 2014 | title = Introduction to Queueing Theory and Stochastic Teletraffic Models | url=http://www.ee.cityu.edu.hk/~zukerman/classnotes.pdf}}
* {{citation | last = Zukerman| first = Moshe| year = 2014| title = Basic Probability Topics | url=http://www.ee.cityu.edu.hk/~zukerman/probability.pdf}}
{{Statistics|state = collapsed}}
{{DEFAULTSORT:Random Variable}}
[[Category:Statistical randomness]]' |
Unified diff of changes made by edit (edit_diff) | '@@ -88,5 +88,5 @@
====Dice roll====
-[[File:Dice Distribution (bar).svg| right | thumb | If the sample space is the set of possible numbers rolled on two dice, and the random variable of interest is the sum ''S'' of the numbers on the two dice, then ''S'' is a discrete random variable whose distribution is described by the [[probability mass function]] plotted as the height of picture columns here.]]
+dodo
A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers ''n''<sub>1</sub> and ''n''<sub>2</sub> from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable ''X'' given by the function that maps the pair to the sum:
' |
New page size (new_size) | 40227 |
Old page size (old_size) | 40589 |
Size change in edit (edit_delta) | -362 |
Lines added in edit (added_lines) | [
0 => 'dodo'
] |
Lines removed in edit (removed_lines) | [
0 => '[[File:Dice Distribution (bar).svg| right | thumb | If the sample space is the set of possible numbers rolled on two dice, and the random variable of interest is the sum ''S'' of the numbers on the two dice, then ''S'' is a discrete random variable whose distribution is described by the [[probability mass function]] plotted as the height of picture columns here.]]'
] |
External links added in the edit (added_links) | [] |
External links removed in the edit (removed_links) | [] |
External links in the page, before the edit (old_links) | [
0 => '//www.ams.org/mathscinet-getitem?mr=0854102',
1 => '//www.ams.org/mathscinet-getitem?mr=0854102',
2 => '//www.worldcat.org/oclc/51441829',
3 => '//www.worldcat.org/oclc/51441829',
4 => 'http://bcs.whfreeman.com/yates2e/',
5 => 'http://econ.ucsb.edu/~doug/245a/Lectures/Measure%20Theory.pdf',
6 => 'http://www.ee.cityu.edu.hk/~zukerman/classnotes.pdf',
7 => 'http://www.ee.cityu.edu.hk/~zukerman/probability.pdf',
8 => 'http://www.mhhe.com/engcs/electrical/papoulis/',
9 => 'http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm',
10 => 'https://books.google.com/?id=L6fhXh13OyMC',
11 => 'https://books.google.com/books/about/A_Modern_Approach_to_Probability_Theory.html?id=5D5O8xyM-kMC',
12 => 'https://books.google.com/books/about/Random_measures.html?id=bBnvAAAAMAAJ',
13 => 'https://books.google.com/books?id=zxXRn-Qmtk8C&pg=PA67',
14 => 'https://mathvault.ca/hub/higher-math/math-symbols/probability-statistics-symbols/',
15 => 'https://web.archive.org/web/20050209001108/http://bcs.whfreeman.com/yates2e/',
16 => 'https://www.encyclopediaofmath.org/index.php?title=Random_variable',
17 => 'https://www.mathsisfun.com/data/random-variables.html'
] |
Parsed HTML source of the new revision (new_html) | '<div class="mw-parser-output"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Variable representing a random phenomenon</div>
<style data-mw-deduplicate="TemplateStyles:r1003042402">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:#f8f9fa;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%}.mw-parser-output .sidebar a{white-space:nowrap}.mw-parser-output .sidebar-wraplinks a{white-space:normal}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding-bottom:0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em 0}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding-top:0.2em;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding-top:0.4em;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.4em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding-top:0}.mw-parser-output .sidebar-image{padding:0.2em 0 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em}.mw-parser-output .sidebar-content{padding:0 0.1em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.4em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%}.mw-parser-output .sidebar-collapse .sidebar-navbar{padding-top:0.6em}.mw-parser-output .sidebar-collapse .mw-collapsible-toggle{margin-top:0.2em}.mw-parser-output .sidebar-list-title{text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}@media(max-width:720px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none;margin-left:0!important;margin-right:0!important}}</style><table class="sidebar vertical-navbox nomobile hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="/wiki/Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Nuvola_apps_atlantik.png" class="image"><img alt="Nuvola apps atlantik.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Nuvola_apps_atlantik.png/100px-Nuvola_apps_atlantik.png" decoding="async" width="100" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/7/77/Nuvola_apps_atlantik.png 1.5x" data-file-width="128" data-file-height="128" /></a></td></tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Probability_axioms" title="Probability axioms">Probability axioms</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li>
<li><a href="/wiki/Sample_space" title="Sample space">Sample space</a></li>
<li><a href="/wiki/Elementary_event" title="Elementary event">Elementary event</a></li>
<li><a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">Event</a></li>
<li><a class="mw-selflink selflink">Random variable</a></li>
<li><a href="/wiki/Probability_measure" title="Probability measure">Probability measure</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Complementary_event" title="Complementary event">Complementary event</a></li>
<li><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li>
<li><a href="/wiki/Marginal_distribution" title="Marginal distribution">Marginal probability</a></li>
<li><a href="/wiki/Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li>
<li><a href="/wiki/Conditional_independence" title="Conditional independence">Conditional independence</a></li>
<li><a href="/wiki/Law_of_total_probability" title="Law of total probability">Law of total probability</a></li>
<li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li>
<li><a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a></li>
<li><a href="/wiki/Boole%27s_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li>
<li><a href="/wiki/Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td>
</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r992953826">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:100%}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_fundamentals" title="Template:Probability fundamentals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_fundamentals" title="Template talk:Probability fundamentals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Template:Probability_fundamentals&action=edit"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table>
<p>In <a href="/wiki/Probability" title="Probability">probability</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, a <b>random variable</b>, <b>random quantity</b>, <b>aleatory variable</b>, or <b>stochastic variable</b> is described informally as a <a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">variable whose values depend</a> on <a href="/wiki/Outcome_(probability)" title="Outcome (probability)">outcomes</a> of a <a href="/wiki/Randomness" title="Randomness">random</a> phenomenon.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup> The formal mathematical treatment of random variables is a topic in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>. In that context, a random variable is understood as a <a href="/wiki/Measurable_function" title="Measurable function">measurable function</a> defined on a <a href="/wiki/Probability_space" title="Probability space">probability space</a> that maps from the <a href="/wiki/Sample_space" title="Sample space">sample space</a> to the <a href="/wiki/Real_number" title="Real number">real numbers</a>.<sup id="cite_ref-UCSB_2-0" class="reference"><a href="#cite_note-UCSB-2">[2]</a></sup>
</p>
<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/wiki/File:Random_Variable_as_a_Function-en.svg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Random_Variable_as_a_Function-en.svg/220px-Random_Variable_as_a_Function-en.svg.png" decoding="async" width="220" height="162" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Random_Variable_as_a_Function-en.svg/330px-Random_Variable_as_a_Function-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Random_Variable_as_a_Function-en.svg/440px-Random_Variable_as_a_Function-en.svg.png 2x" data-file-width="804" data-file-height="592" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Random_Variable_as_a_Function-en.svg" class="internal" title="Enlarge"></a></div>This graph shows how random variable is a function from all possible outcomes to real values. It also shows how random variable is used for defining probability mass functions.</div></div></div>
<p>A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, because of imprecise measurements or <a href="/wiki/Quantum_uncertainty" class="mw-redirect" title="Quantum uncertainty">quantum uncertainty</a>). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but is instead related to philosophical arguments over the <a href="/wiki/Interpretation_of_probability" class="mw-redirect" title="Interpretation of probability">interpretation of probability</a>. The mathematics works the same regardless of the particular interpretation in use.
</p><p>As a function, a random variable is required to be <a href="/wiki/Measurable_function" title="Measurable function">measurable</a>, which allows for probabilities to be assigned to sets of its potential values. It is common that the outcomes depend on some physical variables that are not predictable. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physical conditions, so the outcome being observed is uncertain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.
</p><p>The <a href="/wiki/Domain_of_a_function" title="Domain of a function">___domain</a> of a random variable is called a <i>sample space,</i> defined as the set of possible outcomes of a non-deterministic event. For example, in the event of a coin toss, only two possible outcomes are possible: heads or tails.
</p><p>A random variable has a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a>, which specifies the probability of <a href="/wiki/Borel_subset" class="mw-redirect" title="Borel subset">Borel subsets</a> of its range. Random variables can be <a href="/wiki/Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete</a>, that is, taking any of a specified finite or <a href="/wiki/Countable_set" title="Countable set">countable list</a> of values (having a countable range), endowed with a <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> that is characteristic of the random variable's probability distribution; or <a href="/wiki/Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous</a>, taking any numerical value in an interval or collection of intervals (having an <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable</a> range), via a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> that is characteristic of the random variable's probability distribution; or a mixture of both.
</p><p>Two random variables with the same probability distribution can still differ in terms of their associations with, or <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independence</a> from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called <a href="/wiki/Random_variate" title="Random variate">random variates</a>.
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Definition"><span class="tocnumber">1</span> <span class="toctext">Definition</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Standard_case"><span class="tocnumber">1.1</span> <span class="toctext">Standard case</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="#Extensions"><span class="tocnumber">1.2</span> <span class="toctext">Extensions</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-4"><a href="#Distribution_functions"><span class="tocnumber">2</span> <span class="toctext">Distribution functions</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#Examples"><span class="tocnumber">3</span> <span class="toctext">Examples</span></a>
<ul>
<li class="toclevel-2 tocsection-6"><a href="#Discrete_random_variable"><span class="tocnumber">3.1</span> <span class="toctext">Discrete random variable</span></a>
<ul>
<li class="toclevel-3 tocsection-7"><a href="#Coin_toss"><span class="tocnumber">3.1.1</span> <span class="toctext">Coin toss</span></a></li>
<li class="toclevel-3 tocsection-8"><a href="#Dice_roll"><span class="tocnumber">3.1.2</span> <span class="toctext">Dice roll</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-9"><a href="#Continuous_random_variable"><span class="tocnumber">3.2</span> <span class="toctext">Continuous random variable</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="#Mixed_type"><span class="tocnumber">3.3</span> <span class="toctext">Mixed type</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-11"><a href="#Measure-theoretic_definition"><span class="tocnumber">4</span> <span class="toctext">Measure-theoretic definition</span></a>
<ul>
<li class="toclevel-2 tocsection-12"><a href="#Real-valued_random_variables"><span class="tocnumber">4.1</span> <span class="toctext">Real-valued random variables</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-13"><a href="#Moments"><span class="tocnumber">5</span> <span class="toctext">Moments</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="#Functions_of_random_variables"><span class="tocnumber">6</span> <span class="toctext">Functions of random variables</span></a>
<ul>
<li class="toclevel-2 tocsection-15"><a href="#Example_1"><span class="tocnumber">6.1</span> <span class="toctext">Example 1</span></a></li>
<li class="toclevel-2 tocsection-16"><a href="#Example_2"><span class="tocnumber">6.2</span> <span class="toctext">Example 2</span></a></li>
<li class="toclevel-2 tocsection-17"><a href="#Example_3"><span class="tocnumber">6.3</span> <span class="toctext">Example 3</span></a></li>
<li class="toclevel-2 tocsection-18"><a href="#Example_4"><span class="tocnumber">6.4</span> <span class="toctext">Example 4</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-19"><a href="#Some_properties"><span class="tocnumber">7</span> <span class="toctext">Some properties</span></a></li>
<li class="toclevel-1 tocsection-20"><a href="#Equivalence_of_random_variables"><span class="tocnumber">8</span> <span class="toctext">Equivalence of random variables</span></a>
<ul>
<li class="toclevel-2 tocsection-21"><a href="#Equality_in_distribution"><span class="tocnumber">8.1</span> <span class="toctext">Equality in distribution</span></a></li>
<li class="toclevel-2 tocsection-22"><a href="#Almost_sure_equality"><span class="tocnumber">8.2</span> <span class="toctext">Almost sure equality</span></a></li>
<li class="toclevel-2 tocsection-23"><a href="#Equality"><span class="tocnumber">8.3</span> <span class="toctext">Equality</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-24"><a href="#Convergence"><span class="tocnumber">9</span> <span class="toctext">Convergence</span></a></li>
<li class="toclevel-1 tocsection-25"><a href="#See_also"><span class="tocnumber">10</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-26"><a href="#References"><span class="tocnumber">11</span> <span class="toctext">References</span></a>
<ul>
<li class="toclevel-2 tocsection-27"><a href="#Inline_citations"><span class="tocnumber">11.1</span> <span class="toctext">Inline citations</span></a></li>
<li class="toclevel-2 tocsection-28"><a href="#Literature"><span class="tocnumber">11.2</span> <span class="toctext">Literature</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-29"><a href="#External_links"><span class="tocnumber">12</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<h2><span class="mw-headline" id="Definition">Definition</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=1" title="Edit section: Definition">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>A <b>random variable</b> is a <a href="/wiki/Measurable_function" title="Measurable function">measurable function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \to E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>:<!-- : --></mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \to E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d99cb9d5a88548388921cc682df17876bdecaab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:10.082ex; height:2.176ex;" alt="{\displaystyle X\colon \Omega \to E}"/></span> from a set of possible <a href="/wiki/Outcome_(probability)" title="Outcome (probability)">outcomes</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Omega "/></span> to a <a href="/wiki/Measurable_space" title="Measurable space">measurable space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span>. The technical axiomatic definition requires <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Omega "/></span> to be a sample space of a <a href="/wiki/Probability_space" title="Probability space">probability triple</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
<mo>,</mo>
<mi mathvariant="normal">P</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d59d34e8fd75d8a0c739e420695cc695122073b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.065ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}"/></span> (see the <a href="#Measure-theoretic_definition">measure-theoretic definition</a>). A random variable is often denoted by capital <a href="/wiki/Latin_script" title="Latin script">roman letters</a> such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="Z"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="T"/></span>.<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3">[3]</a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup>
</p><p>The probability that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> takes on a value in a measurable set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>⊆<!-- ⊆ --></mo>
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S\subseteq E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fbdaaaadc900311922bfeaee0de2fa2bc6fe07a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:6.373ex; height:2.343ex;" alt="{\displaystyle S\subseteq E}"/></span> is written as
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} (X\in S)=\operatorname {P} (\{\omega \in \Omega \mid X(\omega )\in S\})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>∈<!-- ∈ --></mo>
<mi>S</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>ω<!-- ω --></mi>
<mo>∈<!-- ∈ --></mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<mi>S</mi>
<mo fence="false" stretchy="false">}</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X\in S)=\operatorname {P} (\{\omega \in \Omega \mid X(\omega )\in S\})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13827ff57e01b13cceff9cf50cd9542cd4b7db70" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:36.004ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (X\in S)=\operatorname {P} (\{\omega \in \Omega \mid X(\omega )\in S\})}"/></span><sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3">[3]</a></sup></dd></dl>
<h3><span class="mw-headline" id="Standard_case">Standard case</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=2" title="Edit section: Standard case">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>In many cases, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> is <a href="/wiki/Real_number" title="Real number">real-valued</a>, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E=\mathbb {R} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/403069620dea86b725997dd3b085172bd11f1bec" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:6.552ex; height:2.176ex;" alt="E = \mathbb{R}"/></span>. In some contexts, the term <a href="/wiki/Random_element" title="Random element">random element</a> (see <a href="#Extensions">extensions</a>) is used to denote a random variable not of this form.
</p><p><span class="anchor" id="Discrete_random_variable"></span>When the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> (or range) of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> is <a href="/wiki/Countable_set" title="Countable set">countable</a>, the random variable is called a <b>discrete random variable</b><sup id="cite_ref-Yates_5-0" class="reference"><a href="#cite_note-Yates-5">[5]</a></sup><sup class="reference" style="white-space:nowrap;">:<span>399</span></sup> and its distribution is a <a href="/wiki/Discrete_probability_distribution" class="mw-redirect" title="Discrete probability distribution">discrete probability distribution</a>, i.e. can be described by a <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> that assigns a probability to each value in the image of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span>. If the image is uncountably infinite (usually an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a>) then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> is called a <b>continuous random variable</b>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6">[6]</a></sup><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Statistics 101 is hardly a sufficient citation, even coming from Yale (October 2020)">citation needed</span></a></i>]</sup> In the special case that it is <a href="/wiki/Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a>, its distribution can be described by a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a>, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7">[7]</a></sup> a <a href="/wiki/Mixture_distribution" title="Mixture distribution">mixture distribution</a> is one such counterexample; such random variables cannot be described by a probability density or a probability mass function.
</p><p>Any random variable can be described by its <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>, which describes the probability that the random variable will be less than or equal to a certain value.
</p>
<h3><span class="mw-headline" id="Extensions">Extensions</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=3" title="Edit section: Extensions">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The term "random variable" in statistics is traditionally limited to the <a href="/wiki/Real_number" title="Real number">real-valued</a> case (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E=\mathbb {R} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/403069620dea86b725997dd3b085172bd11f1bec" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:6.552ex; height:2.176ex;" alt="E=\mathbb {R} "/></span>). In this case, the structure of the real numbers makes it possible to define quantities such as the <a href="/wiki/Expected_value" title="Expected value">expected value</a> and <a href="/wiki/Variance" title="Variance">variance</a> of a random variable, its <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>, and the <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> of its distribution.
</p><p>However, the definition above is valid for any <a href="/wiki/Measurable_space" title="Measurable space">measurable space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span> of values. Thus one can consider random elements of other sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span>, such as random <a href="/wiki/Boolean-valued_function" title="Boolean-valued function">boolean values</a>, <a href="/wiki/Categorical_variable" title="Categorical variable">categorical values</a>, <a href="/wiki/Covariance_matrix#Complex_random_vectors" title="Covariance matrix">complex numbers</a>, <a href="/wiki/Random_vector" class="mw-redirect" title="Random vector">vectors</a>, <a href="/wiki/Random_matrix" title="Random matrix">matrices</a>, <a href="/wiki/Random_sequence" title="Random sequence">sequences</a>, <a href="/wiki/Tree_(graph_theory)" title="Tree (graph theory)">trees</a>, <a href="/wiki/Random_compact_set" title="Random compact set">sets</a>, <a href="/wiki/Shape" title="Shape">shapes</a>, <a href="/wiki/Manifold" title="Manifold">manifolds</a>, and <a href="/wiki/Random_function" class="mw-redirect" title="Random function">functions</a>. One may then specifically refer to a <i>random variable of <a href="/wiki/Data_type" title="Data type">type</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span></i>, or an <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span>-valued random variable</i>.
</p><p>This more general concept of a <a href="/wiki/Random_element" title="Random element">random element</a> is particularly useful in disciplines such as <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, <a href="/wiki/Natural_language_processing" title="Natural language processing">natural language processing</a>, and other fields in <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>, where one is often interested in modeling the random variation of non-numerical <a href="/wiki/Data_structure" title="Data structure">data structures</a>. In some cases, it is nonetheless convenient to represent each element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span>, using one or more real numbers. In this case, a random element may optionally be represented as a <a href="/wiki/Random_vector" class="mw-redirect" title="Random vector">vector of real-valued random variables</a> (all defined on the same underlying probability space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Omega "/></span>, which allows the different random variables to <a href="/wiki/Mutual_information" title="Mutual information">covary</a>). For example:
</p>
<ul><li>A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1\ 0\ 0\ 0\ \cdots )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mtext> </mtext>
<mn>0</mn>
<mtext> </mtext>
<mn>0</mn>
<mtext> </mtext>
<mn>0</mn>
<mtext> </mtext>
<mo>⋯<!-- ⋯ --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (1\ 0\ 0\ 0\ \cdots )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f6e1c3d8f13ec2d8be9bca0eb66a8558367c2e8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.892ex; height:2.843ex;" alt="{\displaystyle (1\ 0\ 0\ 0\ \cdots )}"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0\ 1\ 0\ 0\ \cdots )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mtext> </mtext>
<mn>1</mn>
<mtext> </mtext>
<mn>0</mn>
<mtext> </mtext>
<mn>0</mn>
<mtext> </mtext>
<mo>⋯<!-- ⋯ --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (0\ 1\ 0\ 0\ \cdots )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691482810fe3ec32b87adaa3f47c05fccc580662" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.892ex; height:2.843ex;" alt="{\displaystyle (0\ 1\ 0\ 0\ \cdots )}"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0\ 0\ 1\ 0\ \cdots )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mtext> </mtext>
<mn>0</mn>
<mtext> </mtext>
<mn>1</mn>
<mtext> </mtext>
<mn>0</mn>
<mtext> </mtext>
<mo>⋯<!-- ⋯ --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (0\ 0\ 1\ 0\ \cdots )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee47afca946862178e185c881db6e45df5954927" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.892ex; height:2.843ex;" alt="{\displaystyle (0\ 0\ 1\ 0\ \cdots )}"/></span> and the position of the 1 indicates the word.</li>
<li>A random sentence of given length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="N"/></span> may be represented as a vector of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="N"/></span> random words.</li>
<li>A <a href="/wiki/Random_graph" title="Random graph">random graph</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="N"/></span> given vertices may be represented as a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\times N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
<mo>×<!-- × --></mo>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N\times N}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a86c5231bb3cbb863d9d428ebe9ac8db8d4ffb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:6.968ex; height:2.176ex;" alt="N\times N"/></span> matrix of random variables, whose values specify the <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of the random graph.</li>
<li>A <a href="/wiki/Random_function" class="mw-redirect" title="Random function">random function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="F"/></span> may be represented as a collection of random variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="F(x)"/></span>, giving the function's values at the various points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"/></span> in the function's ___domain. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="F(x)"/></span> are ordinary real-valued random variables provided that the function is real-valued. For example, a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> is a random function of time, a <a href="/wiki/Random_vector" class="mw-redirect" title="Random vector">random vector</a> is a random function of some index set such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,2,\ldots ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1,2,\ldots ,n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e6b15e92183431bb62b787fcdcbdcbe8b40234" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.932ex; height:2.509ex;" alt="{\displaystyle 1,2,\ldots ,n}"/></span>, and <a href="/wiki/Random_field" title="Random field">random field</a> is a random function on any set (typically time, space, or a discrete set).</li></ul>
<h2><span class="mw-headline" id="Distribution_functions">Distribution functions</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=4" title="Edit section: Distribution functions">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>If a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>:<!-- : --></mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68ecb50490003c72b7d6627145965f602b302b2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:9.984ex; height:2.176ex;" alt="X\colon \Omega \to \mathbb {R} "/></span> defined on the probability space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
<mo>,</mo>
<mi mathvariant="normal">P</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d59d34e8fd75d8a0c739e420695cc695122073b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.065ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}"/></span> is given, we can ask questions like "How likely is it that the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> is equal to 2?". This is the same as the probability of the event <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\omega :X(\omega )=2\}\,\!}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>ω<!-- ω --></mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mo fence="false" stretchy="false">}</mo>
<mspace width="thinmathspace" />
<mspace width="negativethinmathspace" />
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{\omega :X(\omega )=2\}\,\!}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cb4fe1d84c37c35452c6c5721382a735981d434" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:15.591ex; height:2.843ex;" alt="\{\omega :X(\omega )=2\}\,\!"/></span> which is often written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X=2)\,\!}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace" />
<mspace width="negativethinmathspace" />
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P(X=2)\,\!}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e44de4aa10f40c4de59aa17c6c607cb3a61e29" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:10.183ex; height:2.843ex;" alt="P(X=2)\,\!"/></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}(2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{X}(2)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5188e81eeb129348060f1b646de0f78570b11539" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.863ex; height:2.843ex;" alt="p_{X}(2)"/></span> for short.
</p><p>Recording all these probabilities of output ranges of a real-valued random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> yields the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span>. The probability distribution "forgets" about the particular probability space used to define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> and only records the probabilities of various values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span>. Such a probability distribution can always be captured by its <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81c05aba576a12b4e05ee3f4cba709dd16139c7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:19.165ex; height:2.843ex;" alt="F_{X}(x)=\operatorname {P} (X\leq x)"/></span></dd></dl>
<p>and sometimes also using a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{X}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc900e2770ed796e420eb5aa0852193a1919ae0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.891ex; height:2.009ex;" alt="p_{X}"/></span>. In <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure-theoretic</a> terms, we use the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> to "push-forward" the measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"/></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Omega "/></span> to a measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{X}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc900e2770ed796e420eb5aa0852193a1919ae0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.891ex; height:2.009ex;" alt="p_{X}"/></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\mathbb {R} "/></span>.
The underlying probability space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Omega "/></span> is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as <a href="/wiki/Correlation_and_dependence" title="Correlation and dependence">correlation and dependence</a> or <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independence</a> based on a <a href="/wiki/Joint_distribution" class="mw-redirect" title="Joint distribution">joint distribution</a> of two or more random variables on the same probability space. In practice, one often disposes of the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Omega "/></span> altogether and just puts a measure on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\mathbb {R} "/></span> that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on <a href="/wiki/Quantile_function" title="Quantile function">quantile functions</a> for fuller development.
</p>
<h2><span class="mw-headline" id="Examples">Examples</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=5" title="Edit section: Examples">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Discrete_random_variable">Discrete random variable</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=6" title="Edit section: Discrete random variable">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>In an experiment a person may be chosen at random, and one random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to the person's height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.
</p><p>Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>PMF</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>PMF</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>PMF</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d91f95b28c52ac8f55c312af264b51ba3bf7b2d6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:35.855ex; height:2.843ex;" alt="{\displaystyle \operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots }"/></span>.
</p><p>In examples such as these, the <a href="/wiki/Sample_space" title="Sample space">sample space</a> is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.
</p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{a_{n}\},\{b_{n}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \{a_{n}\},\{b_{n}\}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82803ec71082ae4fdb9e9821758578b27d8a83d0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.348ex; height:2.843ex;" alt="{\textstyle \{a_{n}\},\{b_{n}\}}"/></span> are countable sets of real numbers, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle b_{n}>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle b_{n}>0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb23ccb34be64412e4d74b40cae449bf0b7ff1fa" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.477ex; height:2.509ex;" alt="{\textstyle b_{n}>0}"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n}b_{n}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munder>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{n}b_{n}=1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c476e636ea5a3f0fc7a7487ef10ee4f47a09a86f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:10.219ex; height:5.509ex;" alt="{\displaystyle \sum _{n}b_{n}=1}"/></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\sum _{n}b_{n}\delta _{a_{n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munder>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F=\sum _{n}b_{n}\delta _{a_{n}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a65b856427f104cbd79423788c927a7f7bcddc08" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:13.897ex; height:5.509ex;" alt="{\displaystyle F=\sum _{n}b_{n}\delta _{a_{n}}}"/></span> is a discrete distribution function. Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{t}(x)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{t}(x)=0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b5c97ea938411c14762eff44de74add8a21f28" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.258ex; height:2.843ex;" alt="{\displaystyle \delta _{t}(x)=0}"/></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo><</mo>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x<t}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d1dc488d4ab095183b700be244e98632e0b5d8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.009ex;" alt="{\displaystyle x<t}"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{t}(x)=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{t}(x)=1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458404a2acdb99c795f12f1bd9c5cb3d516b2efd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.258ex; height:2.843ex;" alt="{\displaystyle \delta _{t}(x)=1}"/></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>≥<!-- ≥ --></mo>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\geq t}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a02705731d972f028007efcdb012b3ac80f954e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle x\geq t}"/></span>. Taking for instance an enumeration of all rational numbers as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="\{a_{n}\}"/></span>, one gets a discrete distribution function that is not a step function or piecewise constant.<sup id="cite_ref-Yates_5-1" class="reference"><a href="#cite_note-Yates-5">[5]</a></sup>
</p>
<h4><span class="mw-headline" id="Coin_toss">Coin toss</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=7" title="Edit section: Coin toss">edit</a><span class="mw-editsection-bracket">]</span></span></h4>
<p>The possible outcomes for one coin toss can be described by the sample space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =\{{\text{heads}},{\text{tails}}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>heads</mtext>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>tails</mtext>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega =\{{\text{heads}},{\text{tails}}\}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/408c39b68acbad53a0195285b2dee7f5978bd967" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:18.109ex; height:2.843ex;" alt="\Omega =\{{\text{heads}},{\text{tails}}\}"/></span>. We can introduce a real-valued random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"/></span> that models a $1 payoff for a successful bet on heads as follows:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(\omega )={\begin{cases}1,&{\text{if }}\omega ={\text{heads}},\\[6pt]0,&{\text{if }}\omega ={\text{tails}}.\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing="0.8em 0.2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mn>1</mn>
<mo>,</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if </mtext>
</mrow>
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>heads</mtext>
</mrow>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
<mo>,</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if </mtext>
</mrow>
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>tails</mtext>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y(\omega )={\begin{cases}1,&{\text{if }}\omega ={\text{heads}},\\[6pt]0,&{\text{if }}\omega ={\text{tails}}.\end{cases}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d200c2fc4177e66d480e649540dd91347a4b0be" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:27.71ex; height:7.509ex;" alt="{\displaystyle Y(\omega )={\begin{cases}1,&{\text{if }}\omega ={\text{heads}},\\[6pt]0,&{\text{if }}\omega ={\text{tails}}.\end{cases}}}"/></span></dd></dl>
<p>If the coin is a <a href="/wiki/Fair_coin" title="Fair coin">fair coin</a>, <i>Y</i> has a <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8b3fed56c8af1f38961f6e4ec0d64fe50ecb4d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.625ex; height:2.509ex;" alt="f_{Y}"/></span> given by:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)={\begin{cases}{\tfrac {1}{2}},&{\text{if }}y=1,\\[6pt]{\tfrac {1}{2}},&{\text{if }}y=0,\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing="0.8em 0.2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mo>,</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mo>,</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)={\begin{cases}{\tfrac {1}{2}},&{\text{if }}y=1,\\[6pt]{\tfrac {1}{2}},&{\text{if }}y=0,\end{cases}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8892016fc9589bd86b7845a45d4882dddbbada" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.671ex; width:24.136ex; height:8.509ex;" alt="{\displaystyle f_{Y}(y)={\begin{cases}{\tfrac {1}{2}},&{\text{if }}y=1,\\[6pt]{\tfrac {1}{2}},&{\text{if }}y=0,\end{cases}}}"/></span></dd></dl>
<h4><span class="mw-headline" id="Dice_roll">Dice roll</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=8" title="Edit section: Dice roll">edit</a><span class="mw-editsection-bracket">]</span></span></h4>
<p>dodo
</p><p>A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers <i>n</i><sub>1</sub> and <i>n</i><sub>2</sub> from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable <i>X</i> given by the function that maps the pair to the sum:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X((n_{1},n_{2}))=n_{1}+n_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X((n_{1},n_{2}))=n_{1}+n_{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63b739139e3242bd25a4aa975f0f9273b8a03924" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:22.367ex; height:2.843ex;" alt="X((n_{1},n_{2}))=n_{1}+n_{2}"/></span></dd></dl>
<p>and (if the dice are <a href="/wiki/Fair_die" class="mw-redirect" title="Fair die">fair</a>) has a probability mass function <i>ƒ</i><sub><i>X</i></sub> given by:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(S)={\frac {\min(S-1,13-S)}{36}},{\text{ for }}S\in \{2,3,4,5,6,7,8,9,10,11,12\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>S</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo movablelimits="true" form="prefix">min</mo>
<mo stretchy="false">(</mo>
<mi>S</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mn>13</mn>
<mo>−<!-- − --></mo>
<mi>S</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>36</mn>
</mfrac>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext> for </mtext>
</mrow>
<mi>S</mi>
<mo>∈<!-- ∈ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo>,</mo>
<mn>4</mn>
<mo>,</mo>
<mn>5</mn>
<mo>,</mo>
<mn>6</mn>
<mo>,</mo>
<mn>7</mn>
<mo>,</mo>
<mn>8</mn>
<mo>,</mo>
<mn>9</mn>
<mo>,</mo>
<mn>10</mn>
<mo>,</mo>
<mn>11</mn>
<mo>,</mo>
<mn>12</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{X}(S)={\frac {\min(S-1,13-S)}{36}},{\text{ for }}S\in \{2,3,4,5,6,7,8,9,10,11,12\}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23da3d676e24e8787b5cdd60a4bbcaf408e82049" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:67.16ex; height:5.676ex;" alt="{\displaystyle f_{X}(S)={\frac {\min(S-1,13-S)}{36}},{\text{ for }}S\in \{2,3,4,5,6,7,8,9,10,11,12\}}"/></span></dd></dl>
<h3><span class="mw-headline" id="Continuous_random_variable">Continuous random variable</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=9" title="Edit section: Continuous random variable">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Formally, a continuous random variable is a random variable whose <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> everywhere.<sup id="cite_ref-:0_8-0" class="reference"><a href="#cite_note-:0-8">[8]</a></sup> There are no "<a href="/wiki/Discontinuity_(mathematics)#Jump_discontinuity" class="mw-redirect" title="Discontinuity (mathematics)">gaps</a>", which would correspond to numbers which have a finite probability of <a href="/wiki/Outcome_(probability)" title="Outcome (probability)">occurring</a>. Instead, continuous random variables <a href="/wiki/Almost_never" class="mw-redirect" title="Almost never">almost never</a> take an exact prescribed value <i>c</i> (formally, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \forall c\in \mathbb {R} :\;\Pr(X=c)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>c</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>:</mo>
<mspace width="thickmathspace" />
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \forall c\in \mathbb {R} :\;\Pr(X=c)=0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ac5d4ef6961bbacd15206d924480ad80a62943" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:24.05ex; height:2.843ex;" alt="{\textstyle \forall c\in \mathbb {R} :\;\Pr(X=c)=0}"/></span>) but there is a positive probability that its value will lie in particular <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a> which can be <a href="/wiki/Arbitrarily_small" class="mw-redirect" title="Arbitrarily small">arbitrarily small</a>. Continuous random variables usually admit <a href="/wiki/Probability_density_function" title="Probability density function">probability density functions</a> (PDF), which characterize their CDF and <a href="/wiki/Probability_measure" title="Probability measure">probability measures</a>;
such distributions are also called <a href="/wiki/Absolutely_continuous_random_variable" class="mw-redirect" title="Absolutely continuous random variable">absolutely continuous</a>; but some continuous distributions are <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a>, or mixes of an absolutely continuous part and a singular part.
</p><p>An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, <i><b>X</b></i> = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any <i>range</i> of values. For example, the probability of choosing a number in [0, 180] is ​<span class="frac nowrap"><sup>1</sup>⁄<sub>2</sub></span>. Instead of speaking of a probability mass function, we say that the probability <i>density</i> of <i><b>X</b></i> is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.
</p><p>More formally, given any <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle I=[a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mi>I</mi>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>:</mo>
<mi>a</mi>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
<mo>≤<!-- ≤ --></mo>
<mi>b</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle I=[a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a291b85e1772a1749a65c841769efcec2931376" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:31.788ex; height:2.843ex;" alt="{\textstyle I=[a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}"/></span>, a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{I}\sim \operatorname {U} (I)=\operatorname {U} [a,b]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msub>
<mo>∼<!-- ∼ --></mo>
<mi mathvariant="normal">U</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">U</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{I}\sim \operatorname {U} (I)=\operatorname {U} [a,b]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e796af5f27ecaec3fdf010383da162a4d29ea04d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.204ex; height:2.843ex;" alt="{\displaystyle X_{I}\sim \operatorname {U} (I)=\operatorname {U} [a,b]}"/></span> is called a "<a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">continuous uniform</a> random variable" (CURV) if the probability that it takes a value in a <a href="/wiki/Subinterval" class="mw-redirect" title="Subinterval">subinterval</a> depends only on the length of the subinterval. This implies that the probability of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{I}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{I}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/064414454fbbf8ad5a6132839c02cf853b1049f9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="X_{I}"/></span> falling in any subinterval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [c,d]\subseteq [a,b]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>c</mi>
<mo>,</mo>
<mi>d</mi>
<mo stretchy="false">]</mo>
<mo>⊆<!-- ⊆ --></mo>
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [c,d]\subseteq [a,b]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c4b90060cfad2522da60caeab608de43226f6e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.204ex; height:2.843ex;" alt="{\displaystyle [c,d]\subseteq [a,b]}"/></span> is <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">length</a> of the subinterval, that is, if <span class="texhtml"><i>a</i> ≤ <i>c</i> ≤ <i>d</i> ≤ <i>b</i></span>, one has
</p><p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr \left(X_{I}\in [c,d]\right)={\frac {d-c}{b-a}}\Pr \left(X_{I}\in I\right)={\frac {d-c}{b-a}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo movablelimits="true" form="prefix">Pr</mo>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<mi>c</mi>
<mo>,</mo>
<mi>d</mi>
<mo stretchy="false">]</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mo>−<!-- − --></mo>
<mi>c</mi>
</mrow>
<mrow>
<mi>b</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mo movablelimits="true" form="prefix">Pr</mo>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mi>I</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mo>−<!-- − --></mo>
<mi>c</mi>
</mrow>
<mrow>
<mi>b</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Pr \left(X_{I}\in [c,d]\right)={\frac {d-c}{b-a}}\Pr \left(X_{I}\in I\right)={\frac {d-c}{b-a}}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb5ff901ae3292273a3c9b43745abceadf82ca9" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.171ex; width:45.147ex; height:5.676ex;" alt="{\displaystyle \Pr \left(X_{I}\in [c,d]\right)={\frac {d-c}{b-a}}\Pr \left(X_{I}\in I\right)={\frac {d-c}{b-a}}}"/></div>
</p><p>where the last equality results from the <a href="/wiki/Probability_axioms#Unitarity" title="Probability axioms">unitarity axiom</a> of probability. The <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> of a CURV <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {U} [a,b]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>∼<!-- ∼ --></mo>
<mi mathvariant="normal">U</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {U} [a,b]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2c5c9c387a71f6b8511c8360740aed05476755" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.377ex; height:2.843ex;" alt="{\displaystyle X\sim \operatorname {U} [a,b]}"/></span> is given by the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> of its interval of <a href="/wiki/Support_(mathematics)" title="Support (mathematics)">support</a> normalized by the interval's length: <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)={\begin{cases}\displaystyle {1 \over b-a},&a\leq x\leq b\\0,&{\text{otherwise}}.\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>b</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mtd>
<mtd>
<mi>a</mi>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
<mo>≤<!-- ≤ --></mo>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
<mo>,</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>otherwise</mtext>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)={\begin{cases}\displaystyle {1 \over b-a},&a\leq x\leq b\\0,&{\text{otherwise}}.\end{cases}}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d5a5660f869d209f7a315d17ccbfeaf0c97801" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -3.505ex; width:30.924ex; height:8.176ex;" alt="{\displaystyle f_{X}(x)={\begin{cases}\displaystyle {1 \over b-a},&a\leq x\leq b\\0,&{\text{otherwise}}.\end{cases}}}"/></div>Of particular interest is the uniform distribution on the <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="[0,1]"/></span>. Samples of any desired <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {D} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">D</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {D} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a9dd309ae777cd525cdf07df0e2b132a8fe6ca" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \operatorname {D} }"/></span> can be generated by calculating the <a href="/wiki/Quantile_function" title="Quantile function">quantile function</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {D} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">D</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {D} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a9dd309ae777cd525cdf07df0e2b132a8fe6ca" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \operatorname {D} }"/></span> on a <a href="/wiki/Random_number_generation" title="Random number generation">randomly-generated number</a> distributed uniformly on the unit interval. This exploits <a href="/wiki/Cumulative_distribution_function#Properties" title="Cumulative distribution function">properties of cumulative distribution functions</a>, which are a unifying framework for all random variables.
</p>
<h3><span class="mw-headline" id="Mixed_type">Mixed type</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=10" title="Edit section: Mixed type">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>A <b>mixed random variable</b> is a random variable whose <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is neither <a href="/wiki/Piecewise_constant" class="mw-redirect" title="Piecewise constant">piecewise-constant</a> (a discrete random variable) nor <a href="/wiki/Continuous_function" title="Continuous function">everywhere-continuous</a>.<sup id="cite_ref-:0_8-1" class="reference"><a href="#cite_note-:0-8">[8]</a></sup> It can be realized as the sum of a discrete random variable and a continuous random variable; in which case the <abbr title="cumulative distribution function">CDF</abbr> will be the weighted average of the CDFs of the component variables.<sup id="cite_ref-:0_8-2" class="reference"><a href="#cite_note-:0-8">[8]</a></sup>
</p><p>An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, <i><b>X</b></i> = −1; otherwise <i><b>X</b></i> = the value of the spinner as in the preceding example. There is a probability of ​<span class="frac nowrap"><sup>1</sup>⁄<sub>2</sub></span> that this random variable will have the value −1. Other ranges of values would have half the probabilities of the last example.
</p><p>Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see <a href="/wiki/Lebesgue%27s_decomposition_theorem#Refinement" title="Lebesgue's decomposition theorem">Lebesgue's decomposition theorem § Refinement</a>. The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
</p>
<h2><span class="mw-headline" id="Measure-theoretic_definition">Measure-theoretic definition</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=11" title="Edit section: Measure-theoretic definition">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>The most formal, <a href="/wiki/Axiomatic" class="mw-redirect" title="Axiomatic">axiomatic</a> definition of a random variable involves <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>. Continuous random variables are defined in terms of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the <a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a>) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">sigma-algebra</a> to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the <a href="/wiki/Borel_%CF%83-algebra" class="mw-redirect" title="Borel σ-algebra">Borel σ-algebra</a>, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> number of <a href="/wiki/Union_(set_theory)" title="Union (set theory)">unions</a> and/or <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersections</a> of such intervals.<sup id="cite_ref-UCSB_2-1" class="reference"><a href="#cite_note-UCSB-2">[2]</a></sup>
</p><p>The measure-theoretic definition is as follows.
</p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
<mo>,</mo>
<mi>P</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="(\Omega ,{\mathcal {F}},P)"/></span> be a <a href="/wiki/Probability_space" title="Probability space">probability space</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/497b309dc3f6f358722b8a00936c8e1ed5c787b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.93ex; height:2.843ex;" alt="(E,{\mathcal {E}})"/></span> a <a href="/wiki/Measurable_space" title="Measurable space">measurable space</a>. Then an <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/497b309dc3f6f358722b8a00936c8e1ed5c787b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.93ex; height:2.843ex;" alt="(E,{\mathcal {E}})"/></span>-valued random variable</b> is a measurable function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \to E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>:<!-- : --></mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \to E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d99cb9d5a88548388921cc682df17876bdecaab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:10.082ex; height:2.176ex;" alt="X\colon \Omega \to E"/></span>, which means that, for every subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {E}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {E}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81710349199c3fc6bcbe14f6758e0c2f1fe9bbca" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.916ex; height:2.176ex;" alt="B\in {\mathcal {E}}"/></span>, its <a href="/wiki/Preimage" class="mw-redirect" title="Preimage">preimage</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{-1}(B)\in {\mathcal {F}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{-1}(B)\in {\mathcal {F}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bed6cfd3c53e4c0affd22f625651158ec8637b8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.67ex; height:3.176ex;" alt="X^{-1}(B)\in {\mathcal {F}}"/></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>ω<!-- ω --></mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<mi>B</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daed213f811c2e8cc77dbb56297b2f8b4688a625" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:26.549ex; height:3.176ex;" alt="X^{-1}(B)=\{\omega :X(\omega )\in B\}"/></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9">[9]</a></sup> This definition enables us to measure any subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {E}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {E}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81710349199c3fc6bcbe14f6758e0c2f1fe9bbca" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.916ex; height:2.176ex;" alt="B\in {\mathcal {E}}"/></span> in the target space by looking at its preimage, which by assumption is measurable.
</p><p>In more intuitive terms, a member of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Omega "/></span> is a possible outcome, a member of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\mathcal {F}}"/></span> is a measurable subset of possible outcomes, the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"/></span> gives the probability of each such measurable subset, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span> represents the set of values that the random variable can take (such as the set of real numbers), and a member of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\mathcal {E}}"/></span> is a "well-behaved" (measurable) subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span> (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.
</p><p>When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span> is a <a href="/wiki/Topological_space" title="Topological space">topological space</a>, then the most common choice for the <a href="/wiki/%CE%A3-algebra" title="Σ-algebra">σ-algebra</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\mathcal {E}}"/></span> is the <a href="/wiki/Borel_%CF%83-algebra" class="mw-redirect" title="Borel σ-algebra">Borel σ-algebra</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(E)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">B</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(E)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3164c0c1413cc8c9b0b9d8b7052d8c7173699f9d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.128ex; height:2.843ex;" alt="{\mathcal {B}}(E)"/></span>, which is the σ-algebra generated by the collection of all open sets in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span>. In such case the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/497b309dc3f6f358722b8a00936c8e1ed5c787b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.93ex; height:2.843ex;" alt="(E,{\mathcal {E}})"/></span>-valued random variable is called an <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span>-valued random variable</b>. Moreover, when the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="E"/></span> is the real line <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\mathbb {R} "/></span>, then such a real-valued random variable is called simply a <b>random variable</b>.
</p>
<h3><span class="mw-headline" id="Real-valued_random_variables">Real-valued random variables</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=12" title="Edit section: Real-valued random variables">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>In this case the observation space is the set of real numbers. Recall, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
<mo>,</mo>
<mi>P</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="(\Omega ,{\mathcal {F}},P)"/></span> is the probability space. For a real observation space, the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \rightarrow \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>:<!-- : --></mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \rightarrow \mathbb {R} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e26dcc35ac81a12c1263c86b05225ef113089fc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:9.984ex; height:2.176ex;" alt="X\colon \Omega \rightarrow \mathbb {R} "/></span> is a real-valued random variable if
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\omega :X(\omega )\leq r\}\in {\mathcal {F}}\qquad \forall r\in \mathbb {R} .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>ω<!-- ω --></mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mi>r</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
<mspace width="2em" />
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>r</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{\omega :X(\omega )\leq r\}\in {\mathcal {F}}\qquad \forall r\in \mathbb {R} .}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/117cd0be4499c834fefa587d5ec055a78f47d178" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:32.009ex; height:2.843ex;" alt="\{\omega :X(\omega )\leq r\}\in {\mathcal {F}}\qquad \forall r\in \mathbb {R} ."/></span></dd></dl>
<p>This definition is a special case of the above because the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(-\infty ,r]:r\in \mathbb {R} \}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">]</mo>
<mo>:</mo>
<mi>r</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{(-\infty ,r]:r\in \mathbb {R} \}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70d40dbc8dd41bbe90d6242042f72e62bafea8f9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.595ex; height:2.843ex;" alt="\{(-\infty ,r]:r\in \mathbb {R} \}"/></span> generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\omega :X(\omega )\leq r\}=X^{-1}((-\infty ,r])}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>ω<!-- ω --></mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mi>r</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{\omega :X(\omega )\leq r\}=X^{-1}((-\infty ,r])}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/967b79350e615a40cee0dd0102fee55bfb3c5d3d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:32.093ex; height:3.176ex;" alt="\{\omega :X(\omega )\leq r\}=X^{-1}((-\infty ,r])"/></span>.
</p>
<h2><span class="mw-headline" id="Moments">Moments</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=13" title="Edit section: Moments">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of <a href="/wiki/Expected_value" title="Expected value">expected value</a> of a random variable, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dd294aa33c0865f58e2b1bdaf44ebe911dbf93" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.857ex; height:2.843ex;" alt="\operatorname {E} [X]"/></span>, and also called the <b>first <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moment</a>.</b> In general, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [f(X)]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">[</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [f(X)]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c407e0dfff7f7d09b8a81f9ccc2f078bffa783ea" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.944ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [f(X)]}"/></span> is not equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\operatorname {E} [X])}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(\operatorname {E} [X])}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/358c53d63b891b58814383d8beba46f69695632f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.944ex; height:2.843ex;" alt="{\displaystyle f(\operatorname {E} [X])}"/></span>. Once the "average value" is known, one could then ask how far from this average value the values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> typically are, a question that is answered by the <a href="/wiki/Variance" title="Variance">variance</a> and <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> of a random variable. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dd294aa33c0865f58e2b1bdaf44ebe911dbf93" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.857ex; height:2.843ex;" alt="\operatorname {E} [X]"/></span> can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span>.
</p><p>Mathematically, this is known as the (generalised) <a href="/wiki/Problem_of_moments" class="mw-redirect" title="Problem of moments">problem of moments</a>: for a given class of random variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span>, find a collection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f_{i}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f_{i}\}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a98b80e62d2a39bb2bdf0dc2c882a3fd4b810c48" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.264ex; height:2.843ex;" alt="\{f_{i}\}"/></span> of functions such that the expectation values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [f_{i}(X)]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">[</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [f_{i}(X)]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e4e5f5f0c5d751d4d1bf63dea54ff9765683a53" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.605ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [f_{i}(X)]}"/></span> fully characterise the <a href="/wiki/Probability_distribution" title="Probability distribution">distribution</a> of the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span>.
</p><p>Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)=X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(X)=X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68563bf25d5ecb2a4dcc2769577eeab1e1ab955" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.146ex; height:2.843ex;" alt="f(X)=X"/></span> of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a <a href="/wiki/Categorical_variable" title="Categorical variable">categorical</a> random variable <i>X</i> that can take on the <a href="/wiki/Nominal_data" class="mw-redirect" title="Nominal data">nominal</a> values "red", "blue" or "green", the real-valued function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X={\text{green}}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>green</mtext>
</mrow>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [X={\text{green}}]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41a3122d8561d29d90be48b6c1fb0f94d8e2a81" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.804ex; height:2.843ex;" alt="[X={\text{green}}]"/></span> can be constructed; this uses the <a href="/wiki/Iverson_bracket" title="Iverson bracket">Iverson bracket</a>, and has the value 1 if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> has the value "green", 0 otherwise. Then, the <a href="/wiki/Expected_value" title="Expected value">expected value</a> and other moments of this function can be determined.
</p>
<h2><span class="mw-headline" id="Functions_of_random_variables">Functions of random variables</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=14" title="Edit section: Functions of random variables">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>A new random variable <i>Y</i> can be defined by <a href="/wiki/Function_composition" title="Function composition">applying</a> a real <a href="/wiki/Measurable_function" title="Measurable function">Borel measurable function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:<!-- : --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae6a1665073bd67dd6b4ef31d23fedad6c62c21" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.12ex; height:2.509ex;" alt="g\colon \mathbb {R} \rightarrow \mathbb {R} "/></span> to the outcomes of a <a href="/wiki/Real-valued" class="mw-redirect" title="Real-valued">real-valued</a> random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span>. That is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=g(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y=g(X)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fd834147d6175f13477acff6567727265d0000" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.777ex; height:2.843ex;" alt="Y=g(X)"/></span>. The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"/></span> is then
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34db583acc468a9227b7a3ef11a8967480f7d75" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:22.242ex; height:2.843ex;" alt="F_{Y}(y)=\operatorname {P} (g(X)\leq y)."/></span></dd></dl>
<p>If function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="g"/></span> is invertible (i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=g^{-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h=g^{-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/224c8e57f10e59865f80b60f807aa160f988792f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.888ex; height:3.009ex;" alt="{\displaystyle h=g^{-1}}"/></span> exists, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="h"/></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="g"/></span>'s <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>) and is either <a href="/wiki/Monotonic_function" title="Monotonic function">increasing or decreasing</a>, then the previous relation can be extended to obtain
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq h(y))=F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq h(y))=1-F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ decreasing}}.\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>≤<!-- ≤ --></mo>
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if </mtext>
</mrow>
<mi>h</mi>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mtext> increasing</mtext>
</mrow>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd />
</mtr>
<mtr>
<mtd>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>≥<!-- ≥ --></mo>
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if </mtext>
</mrow>
<mi>h</mi>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mtext> decreasing</mtext>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq h(y))=F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq h(y))=1-F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ decreasing}}.\end{cases}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda82f7b493f5a2a742ad99a7e5b3e23916b88cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.838ex; width:81.133ex; height:8.843ex;" alt="{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq h(y))=F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq h(y))=1-F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ decreasing}}.\end{cases}}}"/></span></dd></dl>
<p>With the same hypotheses of invertibility of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="g"/></span>, assuming also <a href="/wiki/Differentiability" class="mw-redirect" title="Differentiability">differentiability</a>, the relation between the <a href="/wiki/Probability_density_function" title="Probability density function">probability density functions</a> can be found by differentiating both sides of the above expression with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="y"/></span>, in order to obtain<sup id="cite_ref-:0_8-3" class="reference"><a href="#cite_note-:0-8">[8]</a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)=f_{X}{\bigl (}h(y){\bigr )}\left|{\frac {dh(y)}{dy}}\right|.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.2em" minsize="1.2em">)</mo>
</mrow>
</mrow>
<mrow>
<mo>|</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=f_{X}{\bigl (}h(y){\bigr )}\left|{\frac {dh(y)}{dy}}\right|.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e26c11c24bf65c0792d0abfc5eb9039a9f8a2dae" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:26.964ex; height:6.509ex;" alt="{\displaystyle f_{Y}(y)=f_{X}{\bigl (}h(y){\bigr )}\left|{\frac {dh(y)}{dy}}\right|.}"/></span></dd></dl>
<p>If there is no invertibility of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="g"/></span> but each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="y"/></span> admits at most a countable number of roots (i.e., a finite, or countably infinite, number of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="x_{i}"/></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=g(x_{i})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=g(x_{i})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e8c8917fac0255b472f3fcbb3dfd5349ba2314e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.309ex; height:2.843ex;" alt="{\displaystyle y=g(x_{i})}"/></span>) then the previous relation between the <a href="/wiki/Probability_density_function" title="Probability density function">probability density functions</a> can be generalized with
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow>
<mo>|</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86def9876ca12d7e6c6e073b7b9c408b9ed44c31" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:33.576ex; height:7.176ex;" alt="f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|"/></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=g_{i}^{-1}(y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{i}=g_{i}^{-1}(y)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ae01cf9051bbaa9c5b4e78477edf51930889e6c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:11.644ex; height:3.343ex;" alt="{\displaystyle x_{i}=g_{i}^{-1}(y)}"/></span>, according to the <a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">inverse function theorem</a>. The formulas for densities do not demand <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="g"/></span> to be increasing.
</p><p>In the measure-theoretic, <a href="/wiki/Probability_axioms" title="Probability axioms">axiomatic approach</a> to probability, if a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Omega "/></span> and a <a href="/wiki/Measurable_function" title="Measurable function">Borel measurable function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:<!-- : --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae6a1665073bd67dd6b4ef31d23fedad6c62c21" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.12ex; height:2.509ex;" alt="g\colon \mathbb {R} \rightarrow \mathbb {R} "/></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=g(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y=g(X)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fd834147d6175f13477acff6567727265d0000" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.777ex; height:2.843ex;" alt="{\displaystyle Y=g(X)}"/></span> is also a random variable on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Omega "/></span>, since the composition of measurable functions <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">is also measurable</a>. (However, this is not necessarily true if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="g"/></span> is <a href="/wiki/Lebesgue_measurable" class="mw-redirect" title="Lebesgue measurable">Lebesgue measurable</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2018)">citation needed</span></a></i>]</sup>) The same procedure that allowed one to go from a probability space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,P)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>,</mo>
<mi>P</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\Omega ,P)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c592eb392b832a4e583bbde12cffe5fb5cf975" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.267ex; height:2.843ex;" alt="{\displaystyle (\Omega ,P)}"/></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,dF_{X})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mi>d</mi>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,dF_{X})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ddffbe92024b04bcde1bad6b9468442be4851b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.864ex; height:2.843ex;" alt="(\mathbb {R} ,dF_{X})"/></span> can be used to obtain the distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"/></span>.
</p>
<h3><span class="mw-headline" id="Example_1">Example 1</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=15" title="Edit section: Example 1">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> be a real-valued, <a href="/wiki/Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous random variable</a> and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>=</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y=X^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/164fd6721c56072e02ca8627ae6c18b1efa64e71" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.923ex; height:2.676ex;" alt="{\displaystyle Y=X^{2}}"/></span>.
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Y}(y)=\operatorname {P} (X^{2}\leq y).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>≤<!-- ≤ --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=\operatorname {P} (X^{2}\leq y).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd95dc6913b00a023f868356b416e9894073499b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.388ex; height:3.176ex;" alt="F_{Y}(y)=\operatorname {P} (X^{2}\leq y)."/></span></dd></dl>
<p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo><</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y<0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6a57e5eb282c81c2bb6f5e313012fa77bc08a4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y<0}"/></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X^{2}\leq y)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>≤<!-- ≤ --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P(X^{2}\leq y)=0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24970bfd4938821ee236de96780146c105db2be4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:15.121ex; height:3.176ex;" alt="{\displaystyle P(X^{2}\leq y)=0}"/></span>, so
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Y}(y)=0\qquad {\hbox{if}}\quad y<0.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
<mspace width="2em" />
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mtext>if</mtext>
</mstyle>
</mrow>
<mspace width="1em" />
<mi>y</mi>
<mo><</mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=0\qquad {\hbox{if}}\quad y<0.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcdceed3c03d3e2224bd086e1cf5a5866a17d611" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:24.596ex; height:2.843ex;" alt="F_{Y}(y)=0\qquad {\hbox{if}}\quad y<0."/></span></dd></dl>
<p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\geq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y\geq 0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130e8795bc869a5b823133c5a0972693605c00bd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y\geq 0}"/></span>, then
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} (X^{2}\leq y)=\operatorname {P} (|X|\leq {\sqrt {y}})=\operatorname {P} (-{\sqrt {y}}\leq X\leq {\sqrt {y}}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>≤<!-- ≤ --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mi>X</mi>
<mo>≤<!-- ≤ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X^{2}\leq y)=\operatorname {P} (|X|\leq {\sqrt {y}})=\operatorname {P} (-{\sqrt {y}}\leq X\leq {\sqrt {y}}),}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/647f3a59502b685931f784766844a78fea5645e5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:49.957ex; height:3.509ex;" alt="\operatorname {P} (X^{2}\leq y)=\operatorname {P} (|X|\leq {\sqrt {y}})=\operatorname {P} (-{\sqrt {y}}\leq X\leq {\sqrt {y}}),"/></span></dd></dl>
<p>so
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Y}(y)=F_{X}({\sqrt {y}})-F_{X}(-{\sqrt {y}})\qquad {\hbox{if}}\quad y\geq 0.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
<mo stretchy="false">)</mo>
<mspace width="2em" />
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mtext>if</mtext>
</mstyle>
</mrow>
<mspace width="1em" />
<mi>y</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=F_{X}({\sqrt {y}})-F_{X}(-{\sqrt {y}})\qquad {\hbox{if}}\quad y\geq 0.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4965737980580e870cc77fa21d507b0a13b5bb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:44.137ex; height:3.176ex;" alt="F_{Y}(y)=F_{X}({\sqrt {y}})-F_{X}(-{\sqrt {y}})\qquad {\hbox{if}}\quad y\geq 0."/></span></dd></dl>
<h3><span class="mw-headline" id="Example_2">Example 2</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=16" title="Edit section: Example 2">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> is a random variable with a cumulative distribution
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(x)=P(X\leq x)={\frac {1}{(1+e^{-x})^{\theta }}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=P(X\leq x)={\frac {1}{(1+e^{-x})^{\theta }}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71de30d81204fddf41a7ab8d92bab0e1997d63a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:33.612ex; height:6.009ex;" alt="F_{X}(x)=P(X\leq x)={\frac {1}{(1+e^{-x})^{\theta }}}"/></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta >0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0ac07626379d065418cc158ce6be9aeccf33b9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="\theta >0"/></span> is a fixed parameter. Consider the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\mathrm {log} (1+e^{-X}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">l</mi>
<mi mathvariant="normal">o</mi>
<mi mathvariant="normal">g</mi>
</mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>X</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y=\mathrm {log} (1+e^{-X}).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15eedf0c5e52e3d164e9e67c935cfd91f945fb15" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:18.297ex; height:3.176ex;" alt="{\displaystyle Y=\mathrm {log} (1+e^{-X}).}"/></span> Then,
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Y}(y)=P(Y\leq y)=P(\mathrm {log} (1+e^{-X})\leq y)=P(X\geq -\mathrm {log} (e^{y}-1)).\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>≤<!-- ≤ --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>P</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">l</mi>
<mi mathvariant="normal">o</mi>
<mi mathvariant="normal">g</mi>
</mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>X</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>≥<!-- ≥ --></mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">l</mi>
<mi mathvariant="normal">o</mi>
<mi mathvariant="normal">g</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>.</mo>
<mspace width="thinmathspace" />
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=P(Y\leq y)=P(\mathrm {log} (1+e^{-X})\leq y)=P(X\geq -\mathrm {log} (e^{y}-1)).\,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b546da06b42000fefd3249d4091832619aa90f52" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:67.802ex; height:3.176ex;" alt="{\displaystyle F_{Y}(y)=P(Y\leq y)=P(\mathrm {log} (1+e^{-X})\leq y)=P(X\geq -\mathrm {log} (e^{y}-1)).\,}"/></span></dd></dl>
<p>The last expression can be calculated in terms of the cumulative distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="X,"/></span> so
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F_{Y}(y)&=1-F_{X}(-\log(e^{y}-1))\\[5pt]&=1-{\frac {1}{(1+e^{\log(e^{y}-1)})^{\theta }}}\\[5pt]&=1-{\frac {1}{(1+e^{y}-1)^{\theta }}}\\[5pt]&=1-e^{-y\theta }.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>log</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>log</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>y</mi>
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F_{Y}(y)&=1-F_{X}(-\log(e^{y}-1))\\[5pt]&=1-{\frac {1}{(1+e^{\log(e^{y}-1)})^{\theta }}}\\[5pt]&=1-{\frac {1}{(1+e^{y}-1)^{\theta }}}\\[5pt]&=1-e^{-y\theta }.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4909911022c68c07e09a3cd4722e9c60b62a4f3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -10.671ex; width:31.846ex; height:22.343ex;" alt="{\displaystyle {\begin{aligned}F_{Y}(y)&=1-F_{X}(-\log(e^{y}-1))\\[5pt]&=1-{\frac {1}{(1+e^{\log(e^{y}-1)})^{\theta }}}\\[5pt]&=1-{\frac {1}{(1+e^{y}-1)^{\theta }}}\\[5pt]&=1-e^{-y\theta }.\end{aligned}}}"/></span></dd></dl>
<p>which is the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF) of an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>.
</p>
<h3><span class="mw-headline" id="Example_3">Example 3</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=17" title="Edit section: Example 3">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> is a random variable with a <a href="/wiki/Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal distribution</a>, whose density is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>π<!-- π --></mi>
</msqrt>
</mfrac>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b083a3c3826d10f695ca1b3e66ba8abb567165a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:20.932ex; height:6.176ex;" alt="f_{X}(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}."/></span></dd></dl>
<p>Consider the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>=</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y=X^{2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8194b25626ef14cd837717548e36d3ffebcf2d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.57ex; height:2.676ex;" alt="{\displaystyle Y=X^{2}.}"/></span> We can find the density using the above formula for a change of variables:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow>
<mo>|</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2759b7e26c5b7f286c6ecadec6dbfa550a5400f4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:34.61ex; height:7.176ex;" alt="f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|."/></span></dd></dl>
<p>In this case the change is not <a href="/wiki/Monotonic_function" title="Monotonic function">monotonic</a>, because every value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"/></span> has two corresponding values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)=2f_{X}(g^{-1}(y))\left|{\frac {dg^{-1}(y)}{dy}}\right|.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow>
<mo>|</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=2f_{X}(g^{-1}(y))\left|{\frac {dg^{-1}(y)}{dy}}\right|.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f4674302af1273e9ca28d3115ec5e9d91fcc16" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:32.03ex; height:6.843ex;" alt="f_{Y}(y)=2f_{X}(g^{-1}(y))\left|{\frac {dg^{-1}(y)}{dy}}\right|."/></span></dd></dl>
<p>The inverse transformation is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=g^{-1}(y)={\sqrt {y}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=g^{-1}(y)={\sqrt {y}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa3b24dd2edf00cfbf9641e51b548e0cebd85aa" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:17.034ex; height:3.509ex;" alt="x=g^{-1}(y)={\sqrt {y}}"/></span></dd></dl>
<p>and its derivative is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dg^{-1}(y)}{dy}}={\frac {1}{2{\sqrt {y}}}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {dg^{-1}(y)}{dy}}={\frac {1}{2{\sqrt {y}}}}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c65d4d846f8c011ac2a8021767702285647cd7d6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:17.303ex; height:6.843ex;" alt="{\frac {dg^{-1}(y)}{dy}}={\frac {1}{2{\sqrt {y}}}}."/></span></dd></dl>
<p>Then,
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)=2{\frac {1}{\sqrt {2\pi }}}e^{-y/2}{\frac {1}{2{\sqrt {y}}}}={\frac {1}{\sqrt {2\pi y}}}e^{-y/2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>π<!-- π --></mi>
</msqrt>
</mfrac>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mi>y</mi>
</msqrt>
</mfrac>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=2{\frac {1}{\sqrt {2\pi }}}e^{-y/2}{\frac {1}{2{\sqrt {y}}}}={\frac {1}{\sqrt {2\pi y}}}e^{-y/2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505d29170afe32fcb1823bfffdde7bf35a59e52a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:40.873ex; height:6.509ex;" alt="f_{Y}(y)=2{\frac {1}{\sqrt {2\pi }}}e^{-y/2}{\frac {1}{2{\sqrt {y}}}}={\frac {1}{\sqrt {2\pi y}}}e^{-y/2}."/></span></dd></dl>
<p>This is a <a href="/wiki/Chi-squared_distribution" class="mw-redirect" title="Chi-squared distribution">chi-squared distribution</a> with one <a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">degree of freedom</a>.
</p>
<h3><span class="mw-headline" id="Example_4">Example 4</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=18" title="Edit section: Example 4">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> is a random variable with a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, whose density is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>π<!-- π --></mi>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mfrac>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48359f16b3ba33c0bc3cfe7a70bddd9d1e88c7fc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:29.918ex; height:6.176ex;" alt="{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}"/></span></dd></dl>
<p>Consider the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>=</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y=X^{2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8194b25626ef14cd837717548e36d3ffebcf2d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.57ex; height:2.676ex;" alt="{\displaystyle Y=X^{2}.}"/></span> We can find the density using the above formula for a change of variables:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow>
<mo>|</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2759b7e26c5b7f286c6ecadec6dbfa550a5400f4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:34.61ex; height:7.176ex;" alt="f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|."/></span></dd></dl>
<p>In this case the change is not <a href="/wiki/Monotonic" class="mw-redirect" title="Monotonic">monotonic</a>, because every value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"/></span> has two corresponding values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span> (one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)=f_{X}(g_{1}^{-1}(y))\left|{\frac {dg_{1}^{-1}(y)}{dy}}\right|+f_{X}(g_{2}^{-1}(y))\left|{\frac {dg_{2}^{-1}(y)}{dy}}\right|.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow>
<mo>|</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow>
<mo>|</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=f_{X}(g_{1}^{-1}(y))\left|{\frac {dg_{1}^{-1}(y)}{dy}}\right|+f_{X}(g_{2}^{-1}(y))\left|{\frac {dg_{2}^{-1}(y)}{dy}}\right|.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce25b1a11b8d6f6d6b09a2108360f6ca30a8054" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:54.853ex; height:7.176ex;" alt="{\displaystyle f_{Y}(y)=f_{X}(g_{1}^{-1}(y))\left|{\frac {dg_{1}^{-1}(y)}{dy}}\right|+f_{X}(g_{2}^{-1}(y))\left|{\frac {dg_{2}^{-1}(y)}{dy}}\right|.}"/></span></dd></dl>
<p>The inverse transformation is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=g_{1,2}^{-1}(y)=\pm {\sqrt {y}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=g_{1,2}^{-1}(y)=\pm {\sqrt {y}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2220c9f61ec89ce71a29d3980d1f85f7b7c2c6fd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.338ex; width:18.842ex; height:3.676ex;" alt="{\displaystyle x=g_{1,2}^{-1}(y)=\pm {\sqrt {y}}}"/></span></dd></dl>
<p>and its derivative is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dg_{1,2}^{-1}(y)}{dy}}=\pm {\frac {1}{2{\sqrt {y}}}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msubsup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {dg_{1,2}^{-1}(y)}{dy}}=\pm {\frac {1}{2{\sqrt {y}}}}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b166f940d2cdef6c6afda6a17cb2d39c3e646829" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:19.111ex; height:7.343ex;" alt="{\displaystyle {\frac {dg_{1,2}^{-1}(y)}{dy}}=\pm {\frac {1}{2{\sqrt {y}}}}.}"/></span></dd></dl>
<p>Then,
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}{\frac {1}{2{\sqrt {y}}}}(e^{-({\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}+e^{-(-{\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>π<!-- π --></mi>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>y</mi>
</msqrt>
</mrow>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}{\frac {1}{2{\sqrt {y}}}}(e^{-({\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}+e^{-(-{\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be562627f8e548d04f7a3c9569602926d32c4ae1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:55.718ex; height:6.176ex;" alt="{\displaystyle f_{Y}(y)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}{\frac {1}{2{\sqrt {y}}}}(e^{-({\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}+e^{-(-{\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}).}"/></span></dd></dl>
<p>This is a <a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">noncentral chi-squared distribution</a> with one <a href="/wiki/Degree_of_freedom_(statistics)" class="mw-redirect" title="Degree of freedom (statistics)">degree of freedom</a>.
</p>
<h2><span class="mw-headline" id="Some_properties">Some properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=19" title="Edit section: Some properties">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li>The probability distribution of the sum of two independent random variables is the <b><a href="/wiki/Convolution" title="Convolution">convolution</a></b> of each of their distributions.</li>
<li>Probability distributions are not a <a href="/wiki/Vector_space" title="Vector space">vector space</a>—they are not closed under <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a>, as these do not preserve non-negativity or total integral 1—but they are closed under <a href="/wiki/Convex_combination" title="Convex combination">convex combination</a>, thus forming a <a href="/wiki/Convex_subset" class="mw-redirect" title="Convex subset">convex subset</a> of the space of functions (or measures).</li></ul>
<h2><span class="mw-headline" id="Equivalence_of_random_variables">Equivalence of random variables</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=20" title="Edit section: Equivalence of random variables">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.
</p><p>In increasing order of strength, the precise definition of these notions of equivalence is given below.
</p>
<h3><span class="mw-headline" id="Equality_in_distribution">Equality in distribution</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=21" title="Edit section: Equality in distribution">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>If the sample space is a subset of the real line, random variables <i>X</i> and <i>Y</i> are <i>equal in distribution</i> (denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X{\stackrel {d}{=}}Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-REL">
<mover>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mo>=</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</mover>
</mrow>
</mrow>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X{\stackrel {d}{=}}Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635b086067caf1fb87db7d6670a258111d17b09d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.561ex; height:3.343ex;" alt="X{\stackrel {d}{=}}Y"/></span>) if they have the same distribution functions:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} (X\leq x)=\operatorname {P} (Y\leq x)\quad {\text{for all }}x.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="1em" />
<mrow class="MJX-TeXAtom-ORD">
<mtext>for all </mtext>
</mrow>
<mi>x</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X\leq x)=\operatorname {P} (Y\leq x)\quad {\text{for all }}x.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae0a86fb81da98759d3f4a3887695cf223d644c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:33.195ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (X\leq x)=\operatorname {P} (Y\leq x)\quad {\text{for all }}x.}"/></span></dd></dl>
<p>To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal <a href="/wiki/Moment_generating_function" class="mw-redirect" title="Moment generating function">moment generating functions</a> have the same distribution. This provides, for example, a useful method of checking equality of certain functions of <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent, identically distributed (IID) random variables</a>. However, the moment generating function exists only for distributions that have a defined <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a>.
</p>
<h3><span class="mw-headline" id="Almost_sure_equality">Almost sure equality</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=22" title="Edit section: Almost sure equality">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Two random variables <i>X</i> and <i>Y</i> are <i>equal <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a></i> (denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\;{\stackrel {\text{a.s.}}{=}}\;Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mspace width="thickmathspace" />
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-REL">
<mover>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mo>=</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>a.s.</mtext>
</mrow>
</mover>
</mrow>
</mrow>
<mspace width="thickmathspace" />
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\;{\stackrel {\text{a.s.}}{=}}\;Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2afcd6b106fb5426cd247d7939c13f3281dda5c4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.428ex; height:2.843ex;" alt="{\displaystyle X\;{\stackrel {\text{a.s.}}{=}}\;Y}"/></span>) if, and only if, the probability that they are different is <a href="/wiki/Null_set" title="Null set">zero</a>:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} (X\neq Y)=0.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>≠<!-- ≠ --></mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X\neq Y)=0.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088214863f7b3854c431d827c03f9fd55e017225" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:15.152ex; height:2.843ex;" alt="\operatorname {P} (X\neq Y)=0."/></span></dd></dl>
<p>For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\infty }(X,Y)=\operatorname {ess} \sup _{\omega }|X(\omega )-Y(\omega )|,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>ess</mi>
<mo>⁡<!-- --></mo>
<munder>
<mo movablelimits="true" form="prefix">sup</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ω<!-- ω --></mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>Y</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{\infty }(X,Y)=\operatorname {ess} \sup _{\omega }|X(\omega )-Y(\omega )|,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1a0e3e71662425edd3d71d99c9fb274b2ebdf78" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:34.964ex; height:4.343ex;" alt="{\displaystyle d_{\infty }(X,Y)=\operatorname {ess} \sup _{\omega }|X(\omega )-Y(\omega )|,}"/></span></dd></dl>
<p>where "ess sup" represents the <a href="/wiki/Essential_supremum" class="mw-redirect" title="Essential supremum">essential supremum</a> in the sense of <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>.
</p>
<h3><span class="mw-headline" id="Equality">Equality</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=23" title="Edit section: Equality">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Finally, the two random variables <i>X</i> and <i>Y</i> are <i>equal</i> if they are equal as functions on their measurable space:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(\omega )=Y(\omega )\qquad {\hbox{for all }}\omega .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Y</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mspace width="2em" />
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mtext>for all </mtext>
</mstyle>
</mrow>
<mi>ω<!-- ω --></mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X(\omega )=Y(\omega )\qquad {\hbox{for all }}\omega .}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d39fcd62484d01932de2dc0ea922954927c51c78" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:26.503ex; height:2.843ex;" alt="X(\omega )=Y(\omega )\qquad {\hbox{for all }}\omega ."/></span></dd></dl>
<p>This notion is typically the least useful in probability theory because in practice and in theory, the underlying <a href="/wiki/Measure_space" title="Measure space">measure space</a> of the <a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">experiment</a> is rarely explicitly characterized or even characterizable.
</p>
<h2><span class="mw-headline" id="Convergence">Convergence</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=24" title="Edit section: Convergence">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></div>
<p>A significant theme in mathematical statistics consists of obtaining convergence results for certain <a href="/wiki/Sequence" title="Sequence">sequences</a> of random variables; for instance the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> and the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>.
</p><p>There are various senses in which a sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="X_{n}"/></span> of random variables can converge to a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"/></span>. These are explained in the article on <a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">convergence of random variables</a>.
</p>
<h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=25" title="Edit section: See also">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<style data-mw-deduplicate="TemplateStyles:r936637989">.mw-parser-output .portal{border:solid #aaa 1px;padding:0}.mw-parser-output .portal.tleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portal.tright{margin:0.5em 0 0.5em 1em}.mw-parser-output .portal>ul{display:table;box-sizing:border-box;padding:0.1em;max-width:175px;background:#f9f9f9;font-size:85%;line-height:110%;font-style:italic;font-weight:bold}.mw-parser-output .portal>ul>li{display:table-row}.mw-parser-output .portal>ul>li>span:first-child{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portal>ul>li>span:last-child{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}</style><div role="navigation" aria-label="Portals" class="noprint portal plainlist tright">
<ul>
<li><span><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="image"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="noviewer" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span><span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul></div>
<style data-mw-deduplicate="TemplateStyles:r998391716">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;">
<ul><li><a href="/wiki/Aleatoricism" title="Aleatoricism">Aleatoricism</a></li>
<li><a href="/wiki/Algebra_of_random_variables" title="Algebra of random variables">Algebra of random variables</a></li>
<li><a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">Event (probability theory)</a></li>
<li><a href="/wiki/Multivariate_random_variable" title="Multivariate random variable">Multivariate random variable</a></li>
<li><a href="/wiki/Pairwise_independence" title="Pairwise independence">Pairwise independent random variables</a></li>
<li><a href="/wiki/Observable_variable" title="Observable variable">Observable variable</a></li>
<li><a href="/wiki/Random_element" title="Random element">Random element</a></li>
<li><a href="/wiki/Random_function" class="mw-redirect" title="Random function">Random function</a></li>
<li><a href="/wiki/Random_measure" title="Random measure">Random measure</a></li>
<li><a href="/wiki/Random_number_generator" class="mw-redirect" title="Random number generator">Random number generator</a> produces a random value</li>
<li><a href="/wiki/Random_vector" class="mw-redirect" title="Random vector">Random vector</a></li>
<li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li>
<li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic process</a></li>
<li><a href="/wiki/Relationships_among_probability_distributions" title="Relationships among probability distributions">Relationships among probability distributions</a></li></ul></div>
<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=26" title="Edit section: References">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Inline_citations">Inline citations</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=27" title="Edit section: Inline citations">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<div class="reflist" style="list-style-type: decimal;">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r999302996">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFBlitzsteinHwang2014" class="citation book cs1">Blitzstein, Joe; Hwang, Jessica (2014). <i>Introduction to Probability</i>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781466575592" title="Special:BookSources/9781466575592"><bdi>9781466575592</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Probability&rft.pub=CRC+Press&rft.date=2014&rft.isbn=9781466575592&rft.aulast=Blitzstein&rft.aufirst=Joe&rft.au=Hwang%2C+Jessica&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></span>
</li>
<li id="cite_note-UCSB-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-UCSB_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-UCSB_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFSteigerwald" class="citation web cs1">Steigerwald, Douglas G. <a rel="nofollow" class="external text" href="http://econ.ucsb.edu/~doug/245a/Lectures/Measure%20Theory.pdf">"Economics 245A – Introduction to Measure Theory"</a> <span class="cs1-format">(PDF)</span>. University of California, Santa Barbara<span class="reference-accessdate">. Retrieved <span class="nowrap">April 26,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Economics+245A+%E2%80%93+Introduction+to+Measure+Theory&rft.pub=University+of+California%2C+Santa+Barbara&rft.aulast=Steigerwald&rft.aufirst=Douglas+G.&rft_id=http%3A%2F%2Fecon.ucsb.edu%2F~doug%2F245a%2FLectures%2FMeasure%2520Theory.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></span>
</li>
<li id="cite_note-:1-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathvault.ca/hub/higher-math/math-symbols/probability-statistics-symbols/">"List of Probability and Statistics Symbols"</a>. <i>Math Vault</i>. 2020-04-26<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+Vault&rft.atitle=List+of+Probability+and+Statistics+Symbols&rft.date=2020-04-26&rft_id=https%3A%2F%2Fmathvault.ca%2Fhub%2Fhigher-math%2Fmath-symbols%2Fprobability-statistics-symbols%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/data/random-variables.html">"Random Variables"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Random+Variables&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fdata%2Frandom-variables.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></span>
</li>
<li id="cite_note-Yates-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Yates_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Yates_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFYatesMooreStarnes2003" class="citation book cs1">Yates, Daniel S.; Moore, David S; Starnes, Daren S. (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050209001108/http://bcs.whfreeman.com/yates2e/"><i>The Practice of Statistics</i></a> (2nd ed.). New York: <a href="/wiki/W._H._Freeman_and_Company" title="W. H. Freeman and Company">Freeman</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-4773-4" title="Special:BookSources/978-0-7167-4773-4"><bdi>978-0-7167-4773-4</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://bcs.whfreeman.com/yates2e/">the original</a> on 2005-02-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Practice+of+Statistics&rft.place=New+York&rft.edition=2nd&rft.pub=Freeman&rft.date=2003&rft.isbn=978-0-7167-4773-4&rft.aulast=Yates&rft.aufirst=Daniel+S.&rft.au=Moore%2C+David+S&rft.au=Starnes%2C+Daren+S.&rft_id=http%3A%2F%2Fbcs.whfreeman.com%2Fyates2e%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm">"Random Variables"</a>. <i>www.stat.yale.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.stat.yale.edu&rft.atitle=Random+Variables&rft_id=http%3A%2F%2Fwww.stat.yale.edu%2FCourses%2F1997-98%2F101%2Franvar.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFL._CastañedaV._ArunachalamS._Dharmaraja2012" class="citation book cs1">L. Castañeda; V. Arunachalam & S. Dharmaraja (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zxXRn-Qmtk8C&pg=PA67"><i>Introduction to Probability and Stochastic Processes with Applications</i></a>. Wiley. p. 67. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781118344941" title="Special:BookSources/9781118344941"><bdi>9781118344941</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Probability+and+Stochastic+Processes+with+Applications&rft.pages=67&rft.pub=Wiley&rft.date=2012&rft.isbn=9781118344941&rft.au=L.+Casta%C3%B1eda&rft.au=V.+Arunachalam&rft.au=S.+Dharmaraja&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzxXRn-Qmtk8C%26pg%3DPA67&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></span>
</li>
<li id="cite_note-:0-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_8-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_8-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFBertsekas2002" class="citation book cs1">Bertsekas, Dimitri P. (2002). <i>Introduction to Probability</i>. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/188652940X" title="Special:BookSources/188652940X"><bdi>188652940X</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/51441829">51441829</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Probability&rft.place=Belmont%2C+Mass.&rft.pub=Athena+Scientific&rft.date=2002&rft_id=info%3Aoclcnum%2F51441829&rft.isbn=188652940X&rft.aulast=Bertsekas&rft.aufirst=Dimitri+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></span>
</li>
<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFFristedtGray1996">Fristedt & Gray (1996</a>, page 11)</span>
</li>
</ol></div></div>
<h3><span class="mw-headline" id="Literature">Literature</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=28" title="Edit section: Literature">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<style data-mw-deduplicate="TemplateStyles:r1004355016">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-100{font-size:100%}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns dl,.mw-parser-output .refbegin-columns ol,.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li,.mw-parser-output .refbegin-columns dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="refbegin" style="">
<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFFristedtGray1996" class="citation book cs1">Fristedt, Bert; Gray, Lawrence (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books/about/A_Modern_Approach_to_Probability_Theory.html?id=5D5O8xyM-kMC"><i>A modern approach to probability theory</i></a>. Boston: Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-7643-3807-5" title="Special:BookSources/3-7643-3807-5"><bdi>3-7643-3807-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+modern+approach+to+probability+theory&rft.place=Boston&rft.pub=Birkh%C3%A4user&rft.date=1996&rft.isbn=3-7643-3807-5&rft.aulast=Fristedt&rft.aufirst=Bert&rft.au=Gray%2C+Lawrence&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%2Fabout%2FA_Modern_Approach_to_Probability_Theory.html%3Fid%3D5D5O8xyM-kMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFKallenberg1986" class="citation book cs1"><a href="/wiki/Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, Olav</a> (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books/about/Random_measures.html?id=bBnvAAAAMAAJ"><i>Random Measures</i></a> (4th ed.). Berlin: <a href="/wiki/Akademie_Verlag" title="Akademie Verlag">Akademie Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-394960-2" title="Special:BookSources/0-12-394960-2"><bdi>0-12-394960-2</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0854102">0854102</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Measures&rft.place=Berlin&rft.edition=4th&rft.pub=Akademie+Verlag&rft.date=1986&rft.isbn=0-12-394960-2&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0854102%23id-name%3DMR&rft.aulast=Kallenberg&rft.aufirst=Olav&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%2Fabout%2FRandom_measures.html%3Fid%3DbBnvAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFKallenberg2001" class="citation book cs1">Kallenberg, Olav (2001). <a rel="nofollow" class="external text" href="https://books.google.com/?id=L6fhXh13OyMC"><i>Foundations of Modern Probability</i></a> (2nd ed.). Berlin: <a href="/wiki/Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95313-2" title="Special:BookSources/0-387-95313-2"><bdi>0-387-95313-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Modern+Probability&rft.place=Berlin&rft.edition=2nd&rft.pub=Springer+Verlag&rft.date=2001&rft.isbn=0-387-95313-2&rft.aulast=Kallenberg&rft.aufirst=Olav&rft_id=https%3A%2F%2Fbooks.google.com%2F%3Fid%3DL6fhXh13OyMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFPapoulis1965" class="citation book cs1"><a href="/wiki/Athanasios_Papoulis" title="Athanasios Papoulis">Papoulis, Athanasios</a> (1965). <a rel="nofollow" class="external text" href="http://www.mhhe.com/engcs/electrical/papoulis/"><i>Probability, Random Variables, and Stochastic Processes</i></a> (9th ed.). Tokyo: <a href="/wiki/McGraw%E2%80%93Hill" class="mw-redirect" title="McGraw–Hill">McGraw–Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-119981-0" title="Special:BookSources/0-07-119981-0"><bdi>0-07-119981-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability%2C+Random+Variables%2C+and+Stochastic+Processes&rft.place=Tokyo&rft.edition=9th&rft.pub=McGraw%E2%80%93Hill&rft.date=1965&rft.isbn=0-07-119981-0&rft.aulast=Papoulis&rft.aufirst=Athanasios&rft_id=http%3A%2F%2Fwww.mhhe.com%2Fengcs%2Felectrical%2Fpapoulis%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></li></ul>
</div>
<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_variable&action=edit&section=29" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Random_variable">"Random variable"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Random+variable&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DRandom_variable&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFZukerman2014" class="citation cs2">Zukerman, Moshe (2014), <a rel="nofollow" class="external text" href="http://www.ee.cityu.edu.hk/~zukerman/classnotes.pdf"><i>Introduction to Queueing Theory and Stochastic Teletraffic Models</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Queueing+Theory+and+Stochastic+Teletraffic+Models&rft.date=2014&rft.aulast=Zukerman&rft.aufirst=Moshe&rft_id=http%3A%2F%2Fwww.ee.cityu.edu.hk%2F~zukerman%2Fclassnotes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFZukerman2014" class="citation cs2">Zukerman, Moshe (2014), <a rel="nofollow" class="external text" href="http://www.ee.cityu.edu.hk/~zukerman/probability.pdf"><i>Basic Probability Topics</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Probability+Topics&rft.date=2014&rft.aulast=Zukerman&rft.aufirst=Moshe&rft_id=http%3A%2F%2Fwww.ee.cityu.edu.hk%2F~zukerman%2Fprobability.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+variable" class="Z3988"></span></li></ul>
<div role="navigation" class="navbox" aria-labelledby="Statistics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r992953826"/><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Statistics" title="Template:Statistics"><abbr title="View this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistics" title="Template talk:Statistics"><abbr title="Discuss this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">t</abbr></a></li><li class="nv-edit"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Template:Statistics&action=edit"><abbr title="Edit this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">e</abbr></a></li></ul></div><div id="Statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div id="*_Outline_*_Index">
<ul><li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li>
<li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul>
</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Mean" title="Mean">Mean</a>
<ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic</a></li>
<li><a href="/wiki/Geometric_mean" title="Geometric mean">geometric</a></li>
<li><a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic</a></li></ul></li>
<li><a href="/wiki/Median" title="Median">Median</a></li>
<li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li>
<li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li>
<li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>
<li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li>
<li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li>
<li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>
<li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a>
<ul><li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li>
<li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li>
<li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li>
<li><a href="/wiki/Frequency_distribution" title="Frequency distribution">Frequency distribution</a></li>
<li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" title="Correlation and dependence">Dependence</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>
<li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a>
<ul><li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li>
<li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li></ul></li>
<li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li>
<li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li>
<li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li>
<li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li>
<li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li>
<li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li>
<li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>
<li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li>
<li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li>
<li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li>
<li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li>
<li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li>
<li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li>
<li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>
<li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li>
<li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li>
<li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li>
<li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li>
<li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li>
<li><a href="/wiki/Optimal_design" title="Optimal design">Optimal design</a></li>
<li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li>
<li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li>
<li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>
<ul><li><a href="/wiki/Stratified_sampling" title="Stratified sampling">stratified</a></li>
<li><a href="/wiki/Cluster_sampling" title="Cluster sampling">cluster</a></li></ul></li>
<li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li>
<li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li>
<li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li>
<li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>
<li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>
<li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li>
<li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>
<li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>
<li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive Designs</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Adaptive_clinical_trial" title="Adaptive clinical trial">Adaptive clinical trial</a></li>
<li><a href="/wiki/Up-and-Down_Designs" title="Up-and-Down Designs">Up-and-Down Designs</a></li>
<li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational Studies</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>
<li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li>
<li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li>
<li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>
<li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li>
<li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li>
<li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a>
<ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>
<li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>
<ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>
<li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a>
<ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>
<li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>
<li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a>
<ul><li><a href="/wiki/Location_parameter" title="Location parameter">___location</a></li>
<li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li>
<li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li>
<li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a>
<ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li>
<li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>
<li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>
<li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>
<li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>
<li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>
<ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>
<li><a href="/wiki/U-statistic" title="U-statistic">U</a></li>
<li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li>
<li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a>
<ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li>
<li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>
<li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a>
<ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>
<li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>
<li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a>
<ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>
<li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>
<li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li>
<li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>
<li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>
<ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>
<ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>
<li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>
<li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>
<li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li>
<li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>
<li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>
<li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li>
<li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>
<li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>
<ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>
<li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li>
<li><a href="/wiki/Power_(statistics)" class="mw-redirect" title="Power (statistics)">Power</a>
<ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>
<li><a href="/wiki/Permutation_test" class="mw-redirect" title="Permutation test">Permutation test</a>
<ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>
<li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>
<li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li>
<li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Specific tests</th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:90%;">(normal)</span></a></li>
<li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>
<li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>
<li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li>
<li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>
<li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>
<li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>
<li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>
<li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:90%;">(Shapiro–Wilk)</span></a></li>
<li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>
<li><a href="/wiki/Model_selection" title="Model selection">Model selection</a>
<ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>
<li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li>
<li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a>
<ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>
<li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:90%;">(Wilcoxon)</span></a>
<ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>
<li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:90%;">(Mann–Whitney)</span></a></li>
<li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a>
<ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_one-way_analysis_of_variance" title="Kruskal–Wallis one-way analysis of variance">1-way <span style="font-size:90%;">(Kruskal–Wallis)</span></a></li>
<li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:90%;">(Friedman)</span></a></li>
<li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:90%;">(Jonckheere–Terpstra)</span></a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a>
<ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li>
<li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>
<li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li>
<li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li>
<li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>
<ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist hlist-separated"><ul><li><a href="/wiki/Correlation_and_dependence" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" title="Correlation and dependence">Correlation</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>
<li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li>
<li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li>
<li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">Errors and residuals</a></li>
<li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li>
<li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li>
<li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>
<li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li>
<li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>
<li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li>
<li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>
<li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>
<li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>
<li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li>
<li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li>
<li><a href="/wiki/Heteroscedasticity" title="Heteroscedasticity">Heteroscedasticity</a></li>
<li><a href="/wiki/Homoscedasticity" title="Homoscedasticity">Homoscedasticity</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li>
<li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:90%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>
<li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>
<li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>
<li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li>
<li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li>
<li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li>
<li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li>
<li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li>
<li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>
<li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li>
<li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>
<li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>
<li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>
<li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li>
<li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>
<ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>
<li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>
<ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>
<ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>
<li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>
<li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li>
<li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>
<li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>
<li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li>
<li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li>
<li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>
<li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li>
<li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:90%;">(Ljung–Box)</span></a></li>
<li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>
<li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time ___domain">Time ___domain</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>
<ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>
<li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>
<li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" title="Autoregressive–moving-average model">ARMA model</a></li>
<li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:90%;">(Box–Jenkins)</span></a></li>
<li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>
<li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency ___domain">Frequency ___domain</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>
<li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>
<li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li>
<li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>
<li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>
<li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>
<li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>
<li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li>
<li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li>
<li><a href="/wiki/Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Chemometrics" title="Chemometrics">Chemometrics</a></li>
<li><a href="/wiki/Methods_engineering" title="Methods engineering">Methods engineering</a></li>
<li><a href="/wiki/Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li>
<li><a href="/wiki/Statistical_process_control" title="Statistical process control">Process</a> / <a href="/wiki/Quality_control" title="Quality control">quality control</a></li>
<li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability</a></li>
<li><a href="/wiki/System_identification" title="System identification">System identification</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li>
<li><a href="/wiki/Census" title="Census">Census</a></li>
<li><a href="/wiki/Crime_statistics" title="Crime statistics">Crime statistics</a></li>
<li><a href="/wiki/Demographic_statistics" title="Demographic statistics">Demography</a></li>
<li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li>
<li><a href="/wiki/Jurimetrics" title="Jurimetrics">Jurimetrics</a></li>
<li><a href="/wiki/National_accounts" title="National accounts">National accounts</a></li>
<li><a href="/wiki/Official_statistics" title="Official statistics">Official statistics</a></li>
<li><a href="/wiki/Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li>
<li><a href="/wiki/Psychometrics" title="Psychometrics">Psychometrics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;padding:0px"><div style="padding:0em 0.25em">
<ul><li><a href="/wiki/Cartography" title="Cartography">Cartography</a></li>
<li><a href="/wiki/Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>
<li><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a></li>
<li><a href="/wiki/Geostatistics" title="Geostatistics">Geostatistics</a></li>
<li><a href="/wiki/Kriging" title="Kriging">Kriging</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><img alt="Category" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" title="Category" width="16" height="16" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /><b><a href="/wiki/Category:Statistics" title="Category:Statistics">Category</a></b></li>
<li><b><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="image"><img alt="Nuvola apps edu mathematics blue-p.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="noviewer" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>
<li><img alt="Commons page" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" title="Commons page" width="12" height="16" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw" title="commons:Category:Statistics">Commons</a></b></li>
<li><img alt="WikiProject" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" title="WikiProject" width="16" height="16" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /> <b><a href="/wiki/Wikipedia:WikiProject_Statistics" title="Wikipedia:WikiProject Statistics">WikiProject</a></b></li></ul>
</div></td></tr></tbody></table></div>
' |
Whether or not the change was made through a Tor exit node (tor_exit_node) | false |
Unix timestamp of change (timestamp) | 1613134634 |
External links in the new text (new_links) | [
0 => 'http://econ.ucsb.edu/~doug/245a/Lectures/Measure%20Theory.pdf',
1 => 'https://mathvault.ca/hub/higher-math/math-symbols/probability-statistics-symbols/',
2 => 'https://www.mathsisfun.com/data/random-variables.html',
3 => 'https://web.archive.org/web/20050209001108/http://bcs.whfreeman.com/yates2e/',
4 => 'http://bcs.whfreeman.com/yates2e/',
5 => 'http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm',
6 => 'https://books.google.com/books?id=zxXRn-Qmtk8C&pg=PA67',
7 => '//www.worldcat.org/oclc/51441829',
8 => 'https://books.google.com/books/about/A_Modern_Approach_to_Probability_Theory.html?id=5D5O8xyM-kMC',
9 => 'https://books.google.com/books/about/Random_measures.html?id=bBnvAAAAMAAJ',
10 => '//www.ams.org/mathscinet-getitem?mr=0854102',
11 => 'https://books.google.com/?id=L6fhXh13OyMC',
12 => 'http://www.mhhe.com/engcs/electrical/papoulis/',
13 => 'https://www.encyclopediaofmath.org/index.php?title=Random_variable',
14 => 'http://www.ee.cityu.edu.hk/~zukerman/classnotes.pdf',
15 => 'http://www.ee.cityu.edu.hk/~zukerman/probability.pdf'
] |