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It is absolutely wrong to say an exchangeable sequence is one in which future observations behave like past observations. In particular, that is not the case when they are negatively correlated. |
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In [[statistics]], an '''exchangeable sequence of random variables''' (also sometimes '''interchangeable''')<ref name="ChowTeicher"/> is a sequence
: <math> X_1, X_2, X_3, X_4, X_5, X_6 \text{ and } X_3, X_6, X_1, X_5, X_2, X_4 </math>
both have the same joint probability distribution.
It is closely related to the use of [[independent and identically distributed random variables]] in statistical models. Exchangeable sequences of random variables arise in cases of [[simple random sampling]].
== Definition ==
Formally, an '''exchangeable sequence of random variables''' is a finite or infinite sequence ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ... of [[random variable]]s such that for any finite [[permutation]] σ of the indices 1, 2, 3, ..., (the permutation acts on only finitely many indices, with the rest fixed), the [[joint probability distribution]] of the permuted sequence▼
▲is a finite or infinite sequence ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ... of [[random variable]]s such that for any finite [[permutation]] σ of the indices 1, 2, 3, ..., (the permutation acts on only finitely many indices, with the rest fixed), the [[joint probability distribution]] of the permuted sequence
:<math> X_{\sigma(1)}, X_{\sigma(2)}, X_{\sigma(3)}, \dots</math>
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