# Definition:Random Variable/General Definition

## Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {S, \Sigma'}$ be a measurable space.

A random variable on $\struct {\Omega, \Sigma, \Pr}$, taking values in $\struct {S, \Sigma'}$, is a $\Sigma \, / \, \Sigma'$-measurable mapping $X : \Omega \to S$.

## Notation

As an abuse of notation, we may write:

$\set {\omega \in \Omega : \map P {\map X \omega} }$ as $\set {\map P X}$

for any propositional function of $\map X \omega$ such that:

$\set {\omega \in \Omega : \map P {\map X \omega} }$ is $\Sigma$-measurable.

We may therefore write, for example:

$\map \Pr {\set {\omega \in \Omega : \map X \omega \in B} }$

for some $B \in \Sigma'$, as:

$\map \Pr {\set {X \in B} }$

Usually the curly brackets are dropped and we write:

$\map \Pr {\set {\omega \in \Omega : \map X \omega \in B} } = \map \Pr {X \in B}$

## Also known as

Other words used to mean the same thing as random variable are:

stochastic variable
chance variable
variate.

The image $\Img X$ of $X$ is often denoted $\Omega_X$.