Definition:Random Variable

Informal Definition

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

Formal Definition

General 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 $X : \Omega \to S$.

Real-Valued Random Variable

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

A real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$ is a $\Sigma$-measurable function $X : \Omega \to \R$.

That is, a function $X : \Omega \to \R$ is a real-valued random variable if and only if:

$X^{-1} \sqbrk {\hointl {-\infty} x} = \set {\omega \in \Omega : \map X \omega \le x} \in \Sigma$

for each $x \in \R$, 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 $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

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

A discrete random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$ is a mapping $X: \Omega \to S$ such that:

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

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

Alternatively, the second condition can be written as:

$(2): \quad$ $\forall x \in S: X^{-1} \sqbrk {\set x} \in \Sigma$

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

Continuous Random Variable

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

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$ such that the domain of $X$ is a continuum.

We say that $X$ is a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$ if and only if:

$\ds \lim_{\delta x \mathop \to 0} \map \Pr {X \in \openint x {x + \delta x} } = \map f x \delta x$

where $f$ is the frequency function on $X$.

Absolutely Continuous Random Variable

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

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $P_X$ be the probability distribution of $X$.

Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

We say that $X$ is an absolutely continuous random variable if and only if:

$P_X$ is an absolutely continuous measure with respect to $\lambda$.

Singular Random Variable

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

Let $X$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

We say that $X$ is singular if and only if:

there exists a $\lambda$-null set $B$ such that $\map \Pr {X \in B} = 1$.

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 $\Omega_X$.

Also see

• Results about random variables can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): random variable: 1.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): random variable
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): stochastic variable

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

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