# 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.

### Random Variable

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 $f: \Omega \to X$.

### Random Vector

Let $X_1, X_2, \ldots, X_n$ be random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Then the vector $\mathbf X = \tuple {X_1, X_2, \ldots, X_n}$ is referred to as 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.