Random variable: Difference between revisions

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{{Probability fundamentals}}

In probability and statistics, a '''random variable''', '''random quantity''', '''aleatory variable''', or '''stochastic variable''' is described informally asbasically a [[Dependentmapping andfrom independentset variables|variable whose values depend]] on [[Outcomeof (probability)|outcomes]] of a [[Randomness|random]] phenomenon.<ref>{{cite book|last1=Blitzstein|first1=Joe|last2=Hwang|first2=Jessica|title=Introductionexperiment to Probability|date=2014|publisher=CRCreal Press|isbn=9781466575592}}</ref>numbers. The formal mathematical treatment of random variables is a topic in [[probability theory]]. In that context, a random variable is understood as a [[measurable function]] defined on a [[probability space]] that maps from the [[sample space]] to the real numbers.<ref name="UCSB">{{cite web | title = Economics 245A – Introduction to Measure Theory | url = http://econ.ucsb.edu/~doug/245a/Lectures/Measure%20Theory.pdf | last = Steigerwald | first = Douglas G. | publisher = University of California, Santa Barbara | accessdate = April 26, 2013}}</ref>

[[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.]]

A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, because of imprecise measurements or [[quantum uncertainty]]). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but is instead related to philosophical arguments over the [[interpretation of probability]]. The mathematics works the same regardless of the particular interpretation in use.