Editing Random variable (section)

==Definition==

A '''random variable''' <math>X</math> is a [[measurable function]] <math>X \colon \Omega \to E</math> from a sample space <math> \Omega </math> as a set of possible [[outcome (probability)|outcome]]s to a [[measurable space]] <math> E</math>. The technical axiomatic definition requires the sample space <math>\Omega</math> to belong to 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, Y, Z, T</math>.<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>.

===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 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.

===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).