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

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

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

Alternatively, this condition can be expressed as:
 * $$p_{X} \left({x}\right) = \prod_{k=1}^n p_{X_k} \left({x^k}\right)$$

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 $$i \ne j$$.