Random variable: Difference between revisions

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In the formal mathematical language of [[measure theory]], a random variable is defined as a [[measurable function]] from a [[probability measure space]] (called the ''sample space'') to a [[measurable space]]. This allows consideration of the [[pushforward measure]], which is called the ''distribution'' of the random variable; the distribution is thus a [[probability measure]] on the set of all possible values of the random variable. It is possible for two random variables to have identical distributions but to differ in significant ways; for instance, they may be [[independence (probability theory)|independent]].

It is common to consider the special cases of [[discrete random variable]]s and [[Probability_distribution#Absolutely_continuous_probability_distribution|absolutely continuous random variable]]s, corresponding to whether a random variable is valued in a discretecountable set (such as a finite set)subset or in an interval of [[real number]]s. There are other important possibilities, especially in the theory of [[stochastic process]]es, wherein it is natural to consider [[random sequence]]s or [[random function]]s. Sometimes a ''random variable'' is taken to be automatically valued in the real numbers, with more general random quantities instead being called ''[[random element]]s''.

According to [[George Mackey]], [[Pafnuty Chebyshev]] was the first person "to think systematically in terms of random variables".<ref name=":3">{{cite journal|journal=Bulletin of the American Mathematical Society |series=New Series|volume=3|number=1|date=July 1980|title=Harmonic analysis as the exploitation of symmetry – a historical survey|author=George Mackey}}</ref>

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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 finitely or infinitely [[countable set|countable]], 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}}</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.