# Definition:Random

## Definition

### 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.

## 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:

the image of $X$ is a countable subset of $\Omega'$.

## 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

### 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.

### 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.