# Definition:Random Variable/Definition 2

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## 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:

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

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

## Also see

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 5.1$: Distribution Functions - For a video presentation of the contents of this page, visit the Khan Academy.

- 2001: Michael A. Bean:
*Probability: The Science of Uncertainty*: $\S 2.1$