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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]]
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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'''. This is a technical device used to guarantee the existence of random variables, and sometimes to construct them. In practice, one disposes of the space Ω altogether and just puts a measure on '''R''' that assigns measure 1 to the whole real line.
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