Notation in probability and statistics: Difference between revisions

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==Probability theory==

{{Unreferenced section|date=March 2021}}

* [[Random variable]]s are usually written in [[upper case]] Roman letters, such as <math display="inline">X</math> or <math display="inline">Y</math> and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable. They do not represent a single number or a single category. For instance, if <math>P(X = x) </math> is written, then it represents the probability that a particular realisation of a random variable (e.g., height, number of cars, or bicycle colour), ''X'', would be equal to a particular value or category (e.g., 1.735 m, 52, or purple), <math display="inline">x</math>. It is important that <math display="inline">X</math> and <math display="inline">x</math> are not confused into meaning the same thing. <math display="inline">X</math> is an idea, <math display="inline">x</math> is a value. Clearly they are related, but they do not have identical meanings.

* [[Random variable]]s are usually written in [[upper case]] roman letters: <math display="inline">X</math>, <math display="inline">Y</math>, etc.

* Particular realizationsrealisations of a random variable are written in corresponding [[lower case]] letters. For example, <math display="inline">x_1,x_2, \ldots,x_n</math> could be a [[random sample|sample]] corresponding to the random variable <math display="inline">X</math>. A cumulative probability is formally written <math>P(X\le x) </math> to differentiatedistinguish the random variable from its realization.<ref>{{Cite web |date=2021-08-09 |title=Calculating Probabilities from Cumulative Distribution Function |url=https://analystprep.com/cfa-level-1-exam/quantitative-methods/calculating-probabilities-from-cumulative-distribution-function/ |access-date=2024-02-26}}</ref>

* The probability is sometimes written <math>\mathbb{P} </math> to distinguish it from other functions and measure ''P'' so as to avoid having to define "''P'' is a probability" and <math>\mathbb{P}(X\in A) </math> is short for <math>P(\{\omega \in\Omega: X(\omega) \in A\})</math>, where <math>\Omega</math> is the event space and, <math>X</math> is a random variable that is a function of <math>\omega</math> (i.e., it depends upon <math>\omega</math>), and <math>\omega</math> is some outcome of interest within the ___domain specified by <math>\Omega</math> (say, a randomparticular variableheight, or a particular colour of a car). <math>\Pr(A)</math> notation is used alternatively.

*<math>\mathbb{P}(A \cap B)</math> or <math>\mathbb{P}[B \cap A]</math> indicates the probability that events ''A'' and ''B'' both occur. The [[joint probability distribution]] of random variables ''X'' and ''Y'' is denoted as <math>P(X, Y)</math>, while joint probability mass function or probability density function as <math>f(x, y)</math> and joint cumulative distribution function as <math>F(x, y)</math>.

*<math>\mathbb{P}(A \cup B)</math> or <math>\mathbb{P}[B \cup A]</math> indicates the probability of either event ''A'' or event ''B'' occurring ("or" in this case means [[inclusive or|one or the other or both]]).

*[[sigma-algebra|&sigma;σ-algebras]] are usually written with uppercase [[Calligraphy|calligraphic]] (e.g. <math>\mathcal F</math> for the set of sets on which we define the probability ''P'')

*[[Probability density function]]s (pdfs) and [[probability mass function]]s are denoted by lowercase letters, e.g. <math>f(x)</math>, or <math>f_X(x)</math>.

*[[Cumulative distribution function]]s (cdfs) are denoted by uppercase letters, e.g. <math>F(x)</math>, or <math>F_X(x)</math>.

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*In particular, the pdf of the [[standard normal distribution]] is denoted by <math display="inline">\varphi(z)</math>, and its cdf by <math display="inline">\Phi(z)</math>.

*Some common operators:

:* <math display="inline">\mathrm{E}[X]</math> : [[expected value]] of ''X''

:* <math display="inline">\operatorname{var}[X]</math> : [[variance]] of ''X''

:* <math display="inline">\operatorname{cov}[X,Y]</math> : [[covariance]] of ''X'' and ''Y''

* X is independent of Y is often written <math>X \perp Y</math> or <math>X \perp\!\!\!\perp Y</math>, and X is independent of Y given W is often written

:<math>X \perp\!\!\!\perp Y \,|\, W </math> or

:<math>X \perp Y \,|\, W</math>

* <math>\textstyle P(A\mid B)</math>, the ''[[conditional probability]]'', is the probability of <math>\textstyle A</math> ''given'' <math>\textstyle B</math>, i.e., <mathref>\textstyle{{Citation A</math>|title=Probability ''after''and <math>\textstylestochastic B</math>processes is|date=2013-07-22 observed|url=http://dx.{{factdoi.org/10.1201/b15257-3 |work=Applied Stochastic Processes |pages=9–36 |access-date=May2023-12-08 |publisher=Chapman and Hall/CRC |doi=10.1201/b15257-3 |isbn=978-0-429-16812-3|url-access=subscription 2016}}</ref>

==Statistics==

{{Unreferenced section|date=March 2021}}

*Greek letters (e.g. ''&theta;'', ''&beta;'') are commonly used to denote unknown parameters (population parameters).<ref>{{Cite web |date=1999-02-13 |title=Letters of the Greek Alphabet and Some of Their Statistical Uses |url=https://lesn.appstate.edu/olson/EDL7150/Components/Other%20useful%20links/Greek%20Alphabet%20and%20Statistics.htm |access-date=2024-02-26 |website=les.appstate.edu/}}</ref>

*A tilde (~) denotes "has the probability distribution of".

*Placing a hat, or caret (also known as a circumflex), over a true parameter denotes an [[estimator]] of it, e.g., <math>\widehat{\theta}</math> is an estimator for <math>\theta</math>.

*The [[arithmetic mean]] of a series of values <math display="inline">x_1,x_2, \ldots,x_n</math> is often denoted by placing an "[[overbar]]" over the symbol, e.g. <math>\bar{x}</math>, pronounced "<math display="inline">x</math> bar".

*Some commonly used symbols for [[Sample (statistics)|sample]] statistics are given below:

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**the population [[Pearson product-moment correlation coefficient|correlation]] ''<math display="inline">\rho</math>'',

**the population [[cumulant]]s ''<math display="inline">\kappa_r</math>'',

*<math>x_{(k)}</math> is used for the <math>k^\text{th}</math> [[order statistic]], where <math>x_{(1)}</math> is the sample minimum and <math>x_{(n)}</math> is the sample maximum from a total sample size <math display="inline">n</math>.<ref>{{Cite web |title=Order Statistics |url=https://www.colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf |access-date=2024-02-26 |website=colorado.edu}}</ref>

==Critical values==

{{Unreferenced section|date=March 2021}}

The ''&alpha;α''-level upper [[critical value (statistics)|critical value]] of a [[probability distribution]] is the value exceeded with probability &<math display="inline">\alpha;</math>, that is, the value ''x''<submath display="inline">''&x_\alpha;''</submath> such that ''<math display="inline">F''(''x''_{''&x_\alpha;''}) =&nbsp; 1&nbsp;&minus;&nbsp;''&-\alpha;''</math>, where ''<math display="inline">F''</math> is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

*<math display="inline">z_\alpha</math> or <math display="inline">z(\alpha)</math> for the [[standard normal distribution]]

*<math display="inline">t_{\alpha,\nu}</math> or <math display="inline">t(\alpha,\nu)</math> for the [[Student's t-distribution|''t''-distribution]] with <math display="inline">\nu</math> [[Degrees of freedom (statistics)|degrees of freedom]]