exchangeable distribution

We say the observations $(X_1,\dots ,X_n)$ are exchangeable if for any permutation $\sigma :[n]\to [n]$,

$$(X_1,\dots ,X_n)=(X_{\sigma (1)},\dots ,X_{\sigma (n)})\text{ in distribution.}$$

Exchangeability is clearly a weaker notion than independent and identically distributed. It’s also a strictly stronger notion, as there are distributions that are exchangeable and not iid.

Consider drawing balls from an urn without replacement. The observations are not iid. But they are exchangeable: the probability of drawing red, blue, red is the same as drawing blue, red, blue.