# Definition:Random Variable

## Contents

## Informal Definition

A **random variable** is a number whose value is determined unambiguously by an experiment.

## Formal Definition

### Definition 1

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space, and let $\left({X, \Sigma'}\right)$ be a measurable space.

A **random variable (on $\left({\Omega, \Sigma, \Pr}\right)$)** is a $\Sigma \, / \, \Sigma'$-measurable mapping $f: \Omega \to X$.

### Definition 2

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:

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

### Definition 3

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 unbounded closed interval $\left\{{y \in \R: y \le x}\right\}$;
- $X^{-1} \left({\left({-\infty \,.\,.\, x}\right]}\right)$ denotes the preimage of $\left({-\infty \,.\,.\, x}\right]$ under $X$.

## Discrete Random Variable

Let $\mathcal E$ be an experiment with a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

A **discrete random variable** on $\left({\Omega, \Sigma, \Pr}\right)$ is a mapping $X: \Omega \to \R$ such that:

- $(1): \quad$ The image of $X$ is a countable subset of $\R$
- $(2): \quad$ $\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) = x}\right\} \in \Sigma$

Alternatively (and meaning exactly the same thing), the second condition can be written as:

- $(2)': \quad$ $\forall x \in \R: X^{-1} \left({x}\right) \in \Sigma$

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

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, $\varnothing \in \Sigma$.

Note that a discrete random variable also fulfils the conditions for it to be a random variable.

## Continuous Random Variable

Let $\mathcal E$ be an experiment with a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

A **continuous random variable** on $\left({\Omega, \Sigma, \Pr}\right)$ is a random variable $X: \Omega \to \R$ whose cumulative distribution function is continuous for all $x \in \R$.

## Also known as

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

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

## Sources

- For a video presentation of the contents of this page, visit the Khan Academy.

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