Editing Random variable (section)

{{Short description|Variable representing a random phenomenon}}
{{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 |url=https://www.worldcat.org/oclc/1104219401 |title=Mathematics for machine learning |date=2020 |others=A. Aldo Faisal, Cheng Soon Ong |isbn=978-1-108-47004-9 |location=Cambridge, United Kingdom |oclc=1104219401 |publisher=Cambridge University Press}}</ref> but instead is a mathematical [[function (mathematics)|function]] in which 

* 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
* the [[Range of a function|range]] is a [[measurable space]] (e.g. corresponding to the domain above, the range might be the set <math>\{-1, 1\}</math> if say heads <math>H</math> mapped to -1 and  <math>T</math> mapped to 1). Typically, the range of a random variable is a subset of the [[Real number|real numbers]].

[[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.]]

Informally, randomness typically represents some fundamental element of chance, such as in the roll of a [[dice|die]]; it may also represent uncertainty, such as [[measurement error]].<ref name=":2" /> However, the [[interpretation of probability]] is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous [[Axiom|axiomatic]] setup.

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 countable 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>