Two random variables and are said to be identically distributed if they are defined on the same probability space , and the distribution function of and the distribution function of are the same: . When and are identically distributed, we write .

A set of random variables , in some index set , is identically distributed if for every pair .

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

For example, the interarrival times of a Poisson process of rate are independent and each have an exponential distribution with mean , so the 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.