# Exchangeable Probability Distribution

Let {$Z$} be a measurable space. A probability distribution {$P$} on the measurable space {$Z^n$} of sequences of length {$n$}, where {$n \in {1, 2, \ldots}$}, is exchangeable if

{$P(E) = P\{z_1, \ldots, z_n: z_{\pi(1)}, \ldots, z_{\pi(n)}\in E\}$}

for any measurable {$E \subseteq Z^n$} and any permutation {$\pi$} of the set {$\{1, \ldots, n\}$} (i.e., if the distribution of the sequence {$z_1, \ldots, z_n$} is invariant under any permutation of the indices).

A probability distribution {$P$} on the power measurable space {$Z^{\infty}$} is exchangeable if the marginal distribution {$P_n$} of {$P$} on {$Z^n$} (defined by

{$P_n(E) := P\{(z_1, z_2, \ldots) \in Z^{\infty}: z_1, \ldots, z_n\in E\}$}

for all events {$E \subseteq Z^n$}) is exchangeable for each {$n = 1, 2, \ldots$} (i.e., if the distribution of the sequence {$z_1, z_2, \ldots$} is invariant under any permutation of the finite number of the indices).