Random variable: Difference between revisions

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Mathematically, a random variable is defined as 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]], and we will always assume this in this encyclopedia, unless otherwise specified.

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.

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]].