Random variable: Difference between revisions

Revision as of 01:46, 13 December 2002

Revision as of 22:06, 9 January 2003

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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.

RecordingRecordin all these probabilities of output 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'''.