- Exchangeable random variables
An

**exchangeable sequence of random variables**is asequence "X"_{1}, "X"_{2}, "X"_{3}, ... ofrandom variable s such that for any finitepermutation σ of the indices 1, 2, 3, ..., i.e. any permutation σ that leaves all but finitely many indices fixed, thejoint probability distribution of the permuted sequence:$X\_\{sigma(1)\},\; X\_\{sigma(2)\},\; X\_\{sigma(3)\},\; dots$

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

A sequence "E"

_{1}, "E"_{2}, "E"_{3}, ... of events is said to be exchangeble precisely if the sequence of itsindicator function s is exchangeable.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"}.**Examples*** Any weighted average of

iid sequences of random variables is exchangeble. See in particularde Finetti's theorem .* Suppose an urn contains "n" red and "m" blue marbles. Suppose marbles are drawn without replacement until the urn is empty. Let "X"

_{"i"}be the indicator random variable of the event that the "i"th marble drawn is red. Then {"X"_{"i"}}_{"i"=1,..."n"}is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.* Let "X"

_{1}, "X"_{2}, "X"_{3}, ... be exchangeable random variables, taking real values and such that E("X"_{"i"}^{2}) < ∞. Then E("X"_{1}"X"_{2}) ≥ 0.**ee also***

Hewitt-Savage zero-one law

*de Finetti's theorem **References*** Spizzichino, Fabio "Subjective probability models for lifetimes". Monographs on Statistics and Applied Probability, 91. "Chapman & Hall/CRC", Boca Raton, FL, 2001. xx+248 pp. ISBN 1-58488-060-0

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