Definition:Independent Random Variables

General Definition

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

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

Let $\sequence {X_n}_{n \mathop \in \N}$ be a sequence of random variables on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.

For each $i \in \N$, let $\map \sigma {X_i}$ be the $\sigma$-algebra generated by $X_i$.

We say that $\sequence {X_n}_{n \mathop \in \N}$ is a sequence of independent random variable if and only if:

$\sequence {\map \sigma {X_n} }_{n \mathop \in \N}$ is a sequence of independent $\sigma$-algebras.

Discrete Random Variables

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

Let $X$ and $Y$ be discrete 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:

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

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 discrete random variables.

$X$ 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 $X = \tuple {X_1, X_2, \ldots, X_n}$ be an ordered tuple of discrete 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.