Exchangeable random variables
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An exchangeable sequence of random variables is a sequence X1, X2, X3, ... of random variables such that for any finite permutation σ of the indices 1, 2, 3, ..., i.e. any permutation σ that leaves all but finitely many indices 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 exchangeble precisely if the sequence of its indicator functions is exchangeable.
Independent and identically distributed random variables are exchangeable.
The distribution function FX1,...,Xn(x1, ... ,xn) of a finite sequence of exchangeable random variables is symmetric in its arguments x1, ... ,xn.
[edit] Examples
- Any weighted average of iid sequences of random variables is exchangeble. 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 Xi be the indicator random variable of the event that the ith marble drawn is red. Then {Xi}i=1,...n is an exchangeable sequence. This sequence cannot be extended to any longer exchangeble sequence.
- Let X1, X2, X3, ... be exchangeable random variables, taking real values and such that E(Xi 2) < ∞. Then E(X1 X2) ≥ 0.
[edit] See also
[edit] 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
