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[[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 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]].
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