\begin{definition}[Definition:Random Variable]
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'''.
\end{definition}