Definition:Independent Random Variables

Definition

Let $\mathcal E$ be an experiment with probability space $\left({\Omega, \Sigma, \Pr}\right)$.

Let $X$ and $Y$ be random variables on $\left({\Omega, \Sigma, \Pr}\right)$.

Then $X$ and $Y$ are defined as independent (of each other) iff:

$\Pr \left({X = x, Y = y}\right) = \Pr \left({X = x}\right) \Pr \left({Y = y}\right)$

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

Alternatively, this condition can be expressed as:

$p_{X, Y} \left({x, y}\right) = p_{X} \left({x}\right) p_{Y} \left({y}\right)$

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

$\displaystyle \forall x, y \in \R: p_{X, Y} \left({x, y}\right) = \left({\sum_x p_{X, Y} \left({x, y}\right)}\right) \left({\sum_y p_{X, Y} \left({x, y}\right)}\right)$

General Definition

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

An ordered tuple of random variables $X = \left({X_1, X_1, \ldots, X_n}\right)$ is independent iff:

$\displaystyle \Pr \left({X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n}\right) = \prod_{k=1}^n \Pr \left({X_k = x_k}\right)$

for all $x = \left({x_1, x_2, \ldots, x_n}\right) \in \R^n$.

Alternatively, this condition can be expressed as:

$\displaystyle p_{X} \left({x}\right) = \prod_{k=1}^n p_{X_k} \left({x^k}\right)$

for all $x = \left({x_1, x_2, \ldots, x_n}\right) \in \R^n$.

Pairwise Independent

An ordered tuple of random variables $X = \left({X_1, X_1, \ldots, X_n}\right)$ is pairwise independent iff $X_i$ and $X_j$ are independent whenever $i \ne j$.

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 3.3$