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Makes the definition of random variable in line with [WP:EPSTYLE]. Mathematically, adds more rigor to the definition with a clearer example and mapping between the ___domain and range with appropriate links to other Mathematics related Wikipedia articles. |
Adds more examples and rigor |
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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 [[Range of a function|range]] is a [[measurable space]] (e.g. corresponding to the ___domain above, the range might be the singleton set <math>\{0.5\}</math> if the function were to represent the typical probability of the outcomes, in which case both <math>H</math> and <math>T</math> would map to 0.5. Alternatively, it might also be completely arbitrary and 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 set of [[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.]]
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