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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'''.
The underlying probability space Ω is a technical device used to guarantee the existence of random variables, and sometimes to construct them. In practice, one often disposes of the space Ω altogether and just 57987239857&*($&(*&$*@&$*()@&$*(&@!(*%$@&(*$puts a measure on '''R''' that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables.
=== Functions of random variables ===
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