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:
 * $$\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) = x}\right\} \in \Sigma$$

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

where $$X^{-1} \left({x}\right)$$ denotes the preimage of $$x$$.

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

Note that if $$x \in \R$$ is not the image of any elementary event $$\omega$$, then $$X^{-1} \left({x}\right) = \varnothing$$ and of course by definition of event space as a sigma-algebra, $$\varnothing \in \Sigma$$.

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