exchangeable random variables

A finite setMathworldPlanetmath of random variablesMathworldPlanetmath {X1,,Xn} defined on a common probablility space (Ω,,P) is said to be exchangeable if


for every set of Borel sets {B1,,Bn}, and every permutationMathworldPlanetmath σSn. In other words, X1,,Xn are exchangeable if their joint probability distribution function is the same regardless of their order.

A stochastic processMathworldPlanetmath {Xi} is said to be exchangeable if every finite subset of {Xi} is exchangeable.


  • If S={X1,,Xn} is exchangeable, then every subset of S is exchangeable (by picking suitable Bi and σ). In particular, all Xi are identically distributed, for

  • If S={X1,,Xn} is iid, then S is exchangeable, since the joint distributionPlanetmathPlanetmath of Xi is the productPlanetmathPlanetmath of the distributionsPlanetmathPlanetmathPlanetmath of Xi:

Title exchangeable random variables
Canonical name ExchangeableRandomVariables
Date of creation 2013-03-22 16:25:53
Last modified on 2013-03-22 16:25:53
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 60G09
Synonym exchangeable stochastic process
Defines exchangeable
Defines exchangeable process