You are here
Homeexchangeable random variables
Primary tabs
exchangeable random variables
A finite set of random variables $\{X_{1},\ldots,X_{n}\}$ defined on a common probablility space $(\Omega,\mathcal{F},P)$ is said to be exchangeable if
$P\big((X_{1}\in B_{1})\cap\cdots\cap(X_{n}\in B_{n})\big)=P\big((X_{{\sigma(1)% }}\in B_{1})\cap\cdots\cap(X_{{\sigma(n)}}\in B_{n})\big)$ 
for every set of Borel sets $\{B_{1},\ldots,B_{n}\}$, and every permutation $\sigma\in S_{n}$. In other words, $X_{1},\ldots,X_{n}$ are exchangeable if their joint probability distribution function is the same regardless of their order.
A stochastic process $\{X_{i}\}$ is said to be exchangeable if every finite subset of $\{X_{i}\}$ is exchangeable.
Remarks

If $S=\{X_{1},\ldots,X_{n}\}$ is exchangeable, then every subset of $S$ is exchangeable (by picking suitable $B_{i}$ and $\sigma$). In particular, all $X_{i}$ are identically distributed, for
$P(X_{i}\in B)=P\big((X_{i}\in B)\cap(X_{j}\in\mathbb{R})\big)=P\big((X_{j}\in B% )\cap(X_{i}\in\mathbb{R})\big)=P(X_{j}\in B).$ 
If $S=\{X_{1},\ldots,X_{n}\}$ is iid, then $S$ is exchangeable, since the joint distribution of $X_{i}$ is the product of the distributions of $X_{i}$:
$P\big((X_{1}\in B_{1})\cap\cdots\cap(X_{n}\in B_{n})\big)=P(X_{{\sigma(1)}}\in B% _{1})\big)\cdots P(X_{{\sigma(n)}}\in B_{n})\big).$
Mathematics Subject Classification
60G09 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections