Random variable: Difference between revisions

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We can think of a '''random variable''' as a numeric result of operating a non-deterministic mechanism. Theor mechanismperforming cana benon-deterministic asexperiment. simpleFor asexample, rolling a coin or die toand berecording tossedthe oroutcome yields a rapidrandom countervariable whichwith cyclesrange many{1,2,3,4,5,6}. timesPicking ina therandom intervalperson of aand typicalmeasuring humantheir physicalheight reactionyields timeanother untilrandom stoppedvariable.

Mathematically, wea canrandom describevariable itis defined as a [[measurable function]] whose ___domain isfrom a [[Probability/Sample space|sampleprobability space]] and whose range isto some [[setmeasurable space]] of numbers.

More precisely, it is a [[measurable function]] from a [[probability space]] to some [[measurable space]].

This measurable space is usually taken to be the [[real number]]s with the [[Borel algebra|Borel σ-algebra]].

If a real-valued random variable ''X'', defined on the probability space (Ω, ''P''), is given, we can ask questions like "How likely is it that the value of ''X'' is bigger than 2?". This asks about the probability of the event {''s'' in Ω : ''X''(''s'') > 2} which is often written as ''P''(''X'' > 2) for short.

We can always specify a real-valued random variable by specifying its [[cumulative distribution function]] because two random variables with identical cdf's are isomorphic. From the cdf, we can calculate probabilities for any events which can be described as countable intersections and unions of intervals.

Recording all these probabilities of ouput ranges of a real-valued random variable ''X'' yields the [[probability distribution]] of ''X''. The probability distribution "forgets" about the particular probability space used to define ''X'' and only records the probabilities of various values of ''X''. Such a probability distribution can always be captured by its [[cumulative distribution function]] and sometimes also using a [[probability density function]].

Given a random variable, it is often important to know what its "average value" is. For instance, what is the average you get when you roll a die, or measure a human's height? This is captured by the mathematical concept of [[expected value]] of a random variable.

See also: [[discrete random variable]], [[continuous random variable]], [[probability distribution]]