Exchangeable random variables: Difference between revisions
Appearance
Content deleted Content added
→References: added Aldous |
→Properties: this is wrong without the sigma squared |
||
(2 intermediate revisions by the same user not shown) | |||
Line 20: | Line 20: | ||
== Properties == |
== Properties == |
||
* [[Covariance]]: for exchangeable sequence < |
* [[Covariance]]: for a finite exchangeable sequence { ''X''<sub>''i''</sub> }<sub>''i'' = 1, 2, 3, ...</sub> of length ''n'': |
||
:: <math> \operatorname{Cov} (X_i,X_j) = \text{constant} \ge \frac{-\sigma^2}{n-1},\quad\text{for }i \ne j,</math> |
|||
: where ''σ''<sup> 2</sup> = var(''X''<sub>1</sub>). |
|||
: "Constant" in this case means not depending on the values of the indices ''i'' and ''j'' as long as ''i'' ≠ ''j''. |
|||
* For an infinite exchangeable sequence, |
|||
:: <math> \operatorname{Cov} (X_i,X_j) = \text{constant} \ge 0.\,</math> |
|||
==See also== |
==See also== |
Revision as of 18:18, 13 December 2008
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.
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 exchangeable sequence.
Properties
- Covariance: for a finite exchangeable sequence { Xi }i = 1, 2, 3, ... of length n:
- where σ 2 = var(X1).
- "Constant" in this case means not depending on the values of the indices i and j as long as i ≠ j.
- For an infinite exchangeable sequence,
See also
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 DOI 10.1007/BFb0099421
- 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