Independence of Events and of Random Variables

Let $(\Omega,\mathcal{F},P)$ be a probability space.

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

with the finite product notation. A 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 independent if for all 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 independent and identically distributed (iid) if it is independent and all $X_m$ have the same distribution.