Definition:Random Variable

Informal Definition

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

Formal Definition

Definition 1

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

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

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

Definition 2

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.

A random variable on $\struct {\Omega, \Sigma, \Pr}$ is a mapping $X: \Omega \to \R$ such that:

$\forall x \in \R: \set {\omega \in \Omega: \map X \omega \le x} \in \Sigma$

Definition 3

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

$\forall x \in \R: X^{-1} \sqbrk {\hointl {-\infty} x} \in \Sigma$

where:

$\hointl {-\infty} x$ denotes the unbounded closed interval $\set {y \in \R: y \le x}$

$X^{-1} \sqbrk {\hointl {-\infty} x}$ denotes the preimage of $\hointl {-\infty} x$ under $X$.

Discrete Random Variable

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.

A discrete random variable on $\struct {\Omega, \Sigma, \Pr}$ is a mapping $X: \Omega \to \R$ such that:

$(1): \quad$ The image of $X$ is a countable subset of $\R$

$(2): \quad$ $\forall x \in \R: \set {\omega \in \Omega: \map X \omega = x} \in \Sigma$

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

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

where $\map {X^{-1} } x$ denotes the preimage of $x$.

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

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 $\struct {\Omega, \Sigma, \Pr}$.

A continuous random variable on $\struct {\Omega, \Sigma, \Pr}$ 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 $\Img X$ of $X$ is often denoted $\Omega_X$.

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: random variable

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

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