# Definition:Independent Random Variables

## Definition

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$.

Then $X$ and $Y$ are defined as **independent (of each other)** if and only if:

- $\map \Pr {X = x, Y = y} = \map \Pr {X = x} \map \Pr {Y = y}$

where $\map \Pr {X = x, Y = y}$ is the joint probability mass function of $X$ and $Y$.

Alternatively, this condition can be expressed as:

- $\map {p_{X, Y} } {x, y} = \map {p_X} x \map {p_Y} y$

Using the definition of marginal probability mass function, it can also be expressed as:

- $\displaystyle \forall x, y \in \R: \map {p_{X, Y} } {x, y} = \paren {\sum_x p_{X, Y} \tuple {x, y} } \paren {\sum_y p_{X, Y} \tuple {x, y} }$

### General Definition

The definition can be made to apply to more than just two events.

Let $X = \tuple {X_1, X_1, \ldots, X_n}$ be an ordered tuple of random variables.

### Definition 1

$X$ is **independent** if and only if:

- $\ds \map \Pr {X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n} = \prod_{k \mathop = 1}^n \map \Pr {X_k = x_k}$

for all $x = \tuple {x_1, x_2, \ldots, x_n} \in \R^n$.

### Pairwise Independent

Let $X = \tuple {X_1, X_1, \ldots, X_n}$ be an ordered tuple of random variables.

Then $X$ is **pairwise independent** if and only if $X_i$ and $X_j$ are independent (of each other) whenever $i \ne j$.

## Dependent

Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$.

Then $X$ and $Y$ are defined as **dependent (on each other)** if and only if $X$ and $Y$ are not independent (of each other).

## Also see

- Results about
**independent random variables**can be found here.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 3.3$: Independence of discrete random variables: $(8)$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**independent random variables**