Definition:Random Variable

Definition 1
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space, and let $\left({X, \Sigma'}\right)$ be a measurable space.

A random variable (on $\left({\Omega, \Sigma, \Pr}\right)$) is a $\Sigma \, / \, \Sigma'$-measurable mapping $f: \Omega \to X$.

Definition 2
Let $\mathcal E$ be an experiment with a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

A random variable on $\left({\Omega, \Sigma, \Pr}\right)$ is a mapping $X: \Omega \to \R$ such that:
 * $\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) \le x}\right\} \in \Sigma$

Alternatively (and meaning exactly the same thing), the above condition can be written as:
 * $\forall x \in \R: X^{-1} \left({\left({-\infty \,.\,.\, x}\right]}\right) \in \Sigma$

where:
 * $\left({-\infty \,.\,.\, x}\right]$ denotes the half-open interval $\left\{{y \in \R: y \le x}\right\}$;
 * $X^{-1} \left({\left({-\infty \,.\,.\, x}\right]}\right)$ denotes the preimage of $\left({-\infty \,.\,.\, x}\right]$.

The image $\operatorname{Im} \left({X}\right)$ of $X$ is often denoted $\Omega_X$.

The word variate is often encountered which means the same thing as random variable.

Also see

 * Discrete Random Variable
 * Continuous Random Variable