Exchangeable random variables: Difference between revisions

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An '''exchangeable sequencee of random variables''' is a

Let <math>\xi=\{E_1,\ldots,E_n\}</math> be a family of random events, and let <math>X_1,\ldots,X_n</math> be the [[indicator function]]s of the events in <math>\xi</math>. Then <math>\xi</math> is said to be '''exchangeable''' if, for any [[permutation]]

sequence ''X''₁,&nbsp;''X''₂,&nbsp;''X''₃,&nbsp;... of [[random variable]]s such that for any finite [[permutation]] &sigma; of the indices 1, 2, 3, ..., i.e. any permutation &sigma; that leaves all but finitely many indices fixed, the [[joint probability distribution]] of the permuted sequence

<math>j_1,\ldots,j_n</math> of the indices <math>1,\ldots,n</math>, the two random vectors <math>(X_1,\ldots,X_n)</math> and <math>(X_{j_1},\ldots, X_{j_n})</math> have the same [[joint distribution]].

:<math> X_{\sigma(1)}, X_{\sigma(2)}, X_{\sigma(3)}, \dots</math>

With a more general view, a family of generic random variables <math>(X_1,\ldots,X_n)</math> is exchangeable if, for any permutation <math>j_1\ldots,j_n</math> of the indexes <math>1,\ldots,n</math>, they have the same joint distribution.

is the same as the joint probability distribution of the original sequence.

Independent and identically random variables are exchangeable.

A seqence ''E''₁, ''E''₂, ''E''₃, ... of events is said to be exchangeble precisely if the sequence of its [[indicator function]]s is exchangeable.

An interesting property of exchangeability is that the distribution function <math>F_{X_1,\ldots,X_n}(x_1,\ldots,x_n)</math> is symmetric in its arguments <math>(x_1,\ldots,x_n)</math>.

[[Independent and identically distributed]] random variables are exchangeable.

The distribution function ''F''_{''X''₁,...,''X''_''n''}(''x''₁, ... ,''x''_''n'') of a finite sequence of exchangeable random variables is symmetric in its arguments ''x''₁, ... ,''x''_''n'').

==See also==