Definition:Random Variable

Definition
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.