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Line 1: An '''exchangeable sequence of random variables''' (also sometimes '''interchangeable''') is a sequence ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ... of [[random variable]]s such that for any finite [[permutation]] ~~σ~~σ of the indices 1, 2, 3, ..., i.e. any permutation ~~σ~~σ that leaves all but finitely many indices fixed, the [[joint probability distribution]] of the permuted sequence :<math> X_{\sigma(1)}, X_{\sigma(2)}, X_{\sigma(3)}, \dots</math> Line 6: is the same as the joint probability distribution of the original sequence. A sequence ''E''<sub>1</sub>, ''E''<sub>2</sub>, ''E''<sub>3</sub>, ... of events is said to be ~~exchangeble~~exchangeable precisely if the sequence of its [[indicator function]]s is exchangeable. [[Independent and identically distributed]] random variables are exchangeable. Line 14: == Examples == * Any weighted average of [[iid]] sequences of random variables is ~~exchangeble~~exchangeable. See in particular [[de 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''<sub>''i''</sub> be the indicator random variable of the event that the ''i''th marble drawn is red. Then {''X''<sub>''i''</sub>}<sub>''i''=1,...''n''</sub> is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence. Line 51: == References == * Aldous, David J., ''Exchangeability and related topics'', in: École d'Été de Probabilités de Saint-Flour XIII — 1983, Lecture Notes in Math. 1117, pp. 1-198, Springer, Berlin, 1985. ISBN 978-3-540-15203-3 [http://dx.doi.org/10.1007/BFb0099421 DOI 10.1007/BFb0099421] * Kingman, J. F. C., ''Uses of exchangeability'', Ann. Probability 6 (1978) 83–197 [http://www.ams.org/mathscinet-getitem?mr=494344 MR494344] [http://www.jstor.org/stable/2243211 JSTOR] * Chow, Yuan Shih and Teicher, Henry, ''Probability theory. Independence, interchangeability, martingales,'' Springer Texts in Statistics, 3rd ed., Springer, New York, 1997. xxii+488 pp. ISBN 0-387-98228-0 * 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 [[Category:Probability theory]]

Exchangeable random variables: Difference between revisions