# 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

#### Definition 1

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 $X$.

#### Definition 2

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

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

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {\R, \map \BB \R}$.

Then we say that $X$ is a **real-valued random variable** on $\struct {\Omega, \Sigma, \Pr}$.

## Discrete Random Variable

### Definition 1

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 $\set x$.

### Definition 2

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

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

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.

Then we say that $X$ is a **discrete random variable** on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$ if and only if:

## Continuous Random Variable

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

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

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

- the cumulative distribution function of $X$ is continuous.

### Absolutely Continuous Random Variable

#### Definition 1

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 absolutely continuous with respect to $\lambda$.

#### Definition 2

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

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

Let $F_X$ be the cumulative distribution function of $X$.

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

- $F_X$ is absolutely continuous.

### 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

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.

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