Two random variables $X$ and $Y$ are said to be identically distributed if they are defined on the same probability space $(\Omega,\mathcal{F},P)$ , and the distribution function $F_X$ of $X$ and the distribution function $F_Y$ of $Y$ are the same: $F_X=F_Y$ . When $X$ and $Y$ are identically distributed, we write $X \stackrel{d}{=} Y$ .

A set of random variables $X_i$ , $i$ in some index set $I$ , is identically distributed if $X_i \stackrel{d}{=} X_j$ for every pair $i,j\in I$ .

A collection of random variables $X_i$ ($i\in I$ ) is said to be independent identically distributed, if the $X_i$ 's are identically distributed, and mutually independent (every finite subfamily of $X_i$ is independent). This is often abbreviated as iid.

For example, the interarrival times $T_i$ of a Poisson process of rate $\lambda$ are independent and each have an exponential distribution with mean $1/\lambda$ , so the $T_i$ are independent identically distributed random variables.

Many other examples are found in statistics, where individual data points are often assumed to realizations of iid random variables.