Random variable: Difference between revisions

Revision as of 05:52, 25 April 2002

Revision as of 12:55, 28 April 2002

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If a random variable ''X'':Ω->'''R''' 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 is the same as the probability of the event {''s'' in Ω : ''X''(''s'') > 2} which is often written as ''P''(''X'' > 2) for short.

Recording all these probabilities of ouputoutput 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]]

:''F_X''(x) = P(X≤x)

and sometimes also using a [[probability density function]]. In [[measure theory|measure-theoretic]] terms, we use the random variable ''X'' to "push-forward" the measure ''P'' on Ω to a measure d''F'' on '''R'''.

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Mathematically, this is known as the (generalised) [[problem of moments]]: for a given class of random variables ''X'', find a collection {''f_i''} of functions such that the expectation values E[''f_i''(''X'')] fully characterize the distribution of the random variable ''X''.

See also: [[discrete random variable]], [[continuous random variable]], [[probability distribution]], [[randomness]], [[random vector]], [[random function]], [[generating function]]