Random variable: Difference between revisions

Revision as of 10:04, 25 April 2024

Revision as of 10:23, 25 April 2024

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{{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' can be misleading asin its mathematical definition isrefers notto actuallyneither randomrandomness nor a variable,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 rather itinstead is a mathematical [[function (mathematics)|function]] fromin possiblewhich [[Outcome (probability)|outcomes]] (e.g.,

* the possible upper sides[[Domain of a flippedfunction|___domain]] coinis sucha asset headsof <math>H</math>possible and[[Outcome tails <math>T</math>(probability)|outcomes]] in a [[sample space]] (e.g., the set <math>\{H,T\}</math>) towhich aare [[measurablethe space]]possible (e.g.,upper <math>\{-1,1\}</math>sides inof whicha 1flipped iscoin correspondingheads to <math>H</math> andor −1 is corresponding totails <math>T</math>, respectively),as oftenthe toresult thefrom realtossing numbers.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 singleton set <math>\{0.5\}</math> if the function were to represent the probability of the outcomes). 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.]]