Random variable: Difference between revisions

Revision as of 21:06, 29 June 2019

Revision as of 07:35, 1 July 2019

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'''{{More footnotes|date=February 2012}}

{{Probability fundamentals}}

In [[probability and statistics]], a '''random variable''', '''random quantity''', '''aleatory variable''', or '''stochastic variable''' is described informally as a [[Dependent and independent variables|variable whose values depend]] on [[Outcome (probability)|outcomes]] of a [[Randomness|random]] phenomenon.<ref>{{cite book|last1=Blitzstein|first1=Joe|last2=Hwang|first2=Jessica|title=Introduction to Probability|date=2014|publisher=CRC Press|isbn=9781466575592}}</ref> The formal mathematical treatment of random variables is a topic in [[probability theory]dddddddddd]. In that context, a random variable is understood as a [[measurable function]] defined on a [[sample space]] whose outcomes are typically 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 numerical quantities and also how it 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.

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:<math>d_\infty(X,Y)=\operatorname{ess} \sup_\omega|X(\omega)-Y(\omega)|,</math>

where''''''Bold text'''''''''Bold text'''''Italic text'''''''' "ess sup" represents the [[essential supremum]] in the sense of [[measure theory]].

===Equality===

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{{DEFAULTSORT:Random Variable}}

[[Category:Statistical randomness]]

''''''''Italic text'''''