Definition (Identically Distributed)

A collection of rvs $X_{1},X_{2},\dots$ are said to be identically distributed if for any bounded Borel function $f:\mathbb{R}\to \mathbb{R}$ we have $$\mathbb{E}[f(X_{i})]\text{ does not depend on }i$$which is equivalent to $\mathbb{P}(X_{i}\in E)$ not depending on $i$ for all $E\in \mathcal{B}$.

Definition (iid)

$X_{1},X_{2},\dots$ are iid rv if they’re Independent random variables and .