# Definition:Random

## Definition

### Random Error

### Random Number

### Random Sample

Let $X_i$ be a random variable with $\Img {X_i} = \Omega$, for all $1 \le i \le n$.

Let $F_i$ be the cumulative distribution function of $X_i$ for all $1 \le i \le n$.

We say that $X_1, X_2, \ldots, X_n$ form a **random sample** of size $n$ if:

- $X_i$ and $X_j$ are independent if $i \ne j$
- $\map {F_1} x = \map {F_i} x$ for all $x \in \Omega$

for all $1 \le i, j \le n$.

If $X_1, X_2, \ldots, X_n$ form a **random sample**, they are said to be **independent and identically distributed**, commonly abbreviated **i.i.d**.

### Random Selection

A manner of selecting objects from some larger collection of objects is said to be **random** if the selection is made according to chance.

That is, there is no strict rule or procedure that predictably determines the outcome of the selection.

See experiment and random variable for a precise mathematical treatment of randomness.

### 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*: $\S 2.1$

### Random Vector

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

Let $n \in \N$.

Let $\struct {S_1, \Sigma_1}$, $\struct {S_2, \Sigma_2}$, $\ldots$, $\struct {S_n, \Sigma_n}$ be measurable spaces.

Let:

- $\ds S = \prod_{i \mathop = 1}^n S_i$

For each integer $1 \le i \le n$, let $X_i$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S_i, \Sigma_i}$.

Define a function $\mathbf X : \Omega \to S$ by:

- $\map {\mathbf X} \omega = \tuple {\map {X_1} \omega, \map {X_2} \omega, \ldots, \map {X_n} \omega}$

for each $\omega \in \Omega$.

We call $\mathbf X$ a **random vector**.

### Random Walk

### One-Dimensional Random Walk

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a Markov chain whose state space is the set of integers $\Z$.

Let $\sequence {X_n}$ be such that $X_{n + 1}$ is an element of the set $\set {X_n + 1, X_n, X_n - 1}$.

Then $\sequence {X_n}$ is a **one-dimensional random walk**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**random** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**random**