Exchangeable random variables

In statistics, an exchangeable sequence of random variables (also sometimes interchangeable)^[1] is a sequence X₁, X₂, X₃, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. In other words, the joint distribution is invariant to finite permutation. Thus, for example the sequences

${\displaystyle X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\quad {\text{ and }}\quad X_{3},X_{6},X_{1},X_{5},X_{2},X_{4}}$

both have the same joint probability distribution.

It is closely related to the use of independent and identically distributed random variables in statistical models. Exchangeable sequences of random variables arise in cases of simple random sampling.