Definition:Independent Random Variables/Discrete/General Definition

Definition
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $X = \tuple {X_1, X_1, \ldots, X_n}$ be an ordered tuple of random variables.

Then $X$ is independent :
 * $\displaystyle \map \Pr {X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n} = \prod_{k \mathop = 1}^n \map \Pr {X_k = x_k}$

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

Alternatively, this condition can be expressed as:
 * $\displaystyle \map {p_X} x = \prod_{k \mathop = 1}^n \map {p_{X_k} } {x^k}$

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