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[[Subset]]{{Short description|Variable representing a random phenomenon}}
{{Probability fundamentals}}
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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 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>
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