Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}.

Events $A_1,\dots,A_r\in\mathcal{F}$ are \textbf{independent} if for every nonempty subset $S\subseteq\{1,\dots,r\}$,

with the \reftext{def:finite-product-notation-2026a}{finite product notation}. A \reftext{def:sequence-in-set-2026a}{sequence} (or arbitrary family) of events is independent if every finite subfamily is independent.

Random variables $X_1,\dots,X_r$ on $(\Omega,\mathcal{F},P)$ are \textbf{independent} if for all \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel sets} $B_1,\dots,B_r$ the events $\{X_1\in B_1\},\dots,\{X_r\in B_r\}$ are independent. A sequence (or family) of random variables is independent if every finite subfamily is independent.

A sequence $(X_m)_{m\in\mathbb{N}}$ of random variables is \textbf{independent and identically distributed} (\textbf{iid}) if it is independent and all $X_m$ have the same \reftext{def:distribution-cdf-random-variable-2026a}{distribution}.