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{{Probability fundamentals}}
A '''random variable''' (also called '''random quantity''', '''aleatory variable''', or '''stochastic variable''') is a [[Mathematics| mathematical]] formalization of a quantity or object which depends on [[randomness|random]] events.<ref name=":2">{{cite book|last1=Blitzstein|first1=Joe|title=Introduction to Probability|last2=Hwang|first2=Jessica|date=2014|publisher=CRC Press|isbn=9781466575592}}</ref> The term 'random variable' in its mathematical definition refers to neither randomness nor variability<ref>{{Cite book |last=Deisenroth |first=Marc Peter
* the [[Domain of a function|___domain]] is the set of possible [[Outcome (probability)|outcomes]] in a [[sample space]] (e.g. the set <math>\{H,T\}</math> which are the possible upper sides of a flipped coin heads <math>H</math> or tails <math>T</math> as the result from tossing a coin); and
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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 finite or [[countable set|countably]] infinite, the random variable is called a '''discrete random variable'''<ref name="Yates">{{cite book | last1 = Yates | first1 = 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><ref>{{Cite journal|last1=Dekking|first1=Frederik Michel|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|date=2005|title=A Modern Introduction to Probability and Statistics|url=https://doi.org/10.1007/1-84628-168-7|journal=Springer Texts in Statistics|language=en-gb|doi=10.1007/1-84628-168-7|isbn=978-1-85233-896-1|issn=1431-875X|url-access=subscription}}</ref> 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>
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.
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*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==
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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 display="inline">\sum_n b_n=1</math>, then <math display="inline"> F=\sum_n b_n \delta_{a_n}(x)</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 function that is not necessarily a [[step function]] ([[piecewise]] constant).
====Coin toss====
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