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Concept# Exchangeable random variables

Summary

In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. Thus, for example the sequences
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.
Formally, an exchangeable sequence of random variables is a finite or infinite sequence X1, X2, X3, ... of random variables 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 the same as the joint probability distribution of the original sequence.
(A sequence E1, E2, E3, ... of events is said to be exchangeable precisely if the sequence of its indicator functions is exchangeable.) The distribution function FX1,...,Xn(x1, ..., xn) of a finite sequence of exchangeable random variables is symmetric in its arguments x1, ..., xn. Olav Kallenberg provided an appropriate definition of exchangeability for continuous-time stochastic processes.
The concept was introduced by William Ernest Johnson in his 1924 book Logic, Part III: The Logical Foundations of Science. Exchangeability is equivalent to the concept of statistical control introduced by Walter Shewhart also in 1924.
The property of exchangeability is closely related to the use of independent and identically distributed (i.i.d.) random variables in statistical models. A sequence of random variables that are i.i.d, conditional on some underlying distributional form, is exchangeable. This follows directly from the structure of the joint probability distribution generated by the i.i.d. form.
Mixtures of exchangeable sequences (in particular, sequences of i.

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