Definition:Independent Random Variables/General Definition

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

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

Then $X$ is independent iff:

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

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

Alternatively, this condition can be expressed as:

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

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