Exchangeable random variables


Exchangeable random variables

An exchangeable sequence of random variables is asequence "X"1, "X"2, "X"3, ... 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

: X_{sigma(1)}, X_{sigma(2)}, X_{sigma(3)}, dots

is the same as the joint probability distribution of the original sequence.

A sequence "E"1, "E"2, "E"3, ... 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 "F""X"1,...,"X""n"("x"1, ... ,"x""n") of a finite sequence of exchangeable random variables is symmetric in its arguments "x"1, ... ,"x""n".

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 "X""i" be the indicator random variable of the event that the "i"th marble drawn is red. Then {"X""i"}"i"=1,..."n" is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.

* Let "X"1, "X"2, "X"3, ... be exchangeable random variables, taking real values and such that E("X""i" 2) < ∞. Then E("X"1 "X"2) ≥ 0.

ee also

* Hewitt-Savage zero-one law
* de Finetti's theorem

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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • de Finetti's theorem — In probability theory, de Finetti s theorem explains why exchangeable observations are conditionally independent given some latent variable to which an epistemic probability distribution would then be assigned. It is named in honor of Bruno de… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • List of mathematics articles (E) — NOTOC E E₇ E (mathematical constant) E function E₈ lattice E₈ manifold E∞ operad E7½ E8 investigation tool Earley parser Early stopping Earnshaw s theorem Earth mover s distance East Journal on Approximations Eastern Arabic numerals Easton s… …   Wikipedia

  • De Finetti's theorem — In probability theory, de Finetti s theorem explains why exchangeable observations are conditionally independent given some (usually) unobservable quantity to which an epistemic probability distribution would then be assigned. It is named in… …   Wikipedia

  • Dirichlet process — In probability theory, a Dirichlet process is a stochastic process that can be thought of as a probability distribution whose domain is itself a random distribution. That is, given a Dirichlet process , where H (the base distribution) is an… …   Wikipedia

  • Stein's method — is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it 1972,[1] to obtain a bound between… …   Wikipedia

  • Hewitt-Savage zero-one law — The Hewitt Savage zero one law is a theorem in probability theory, similar to Kolmogorov s zero one law, that specifies that a certain type of event will either almost surely happen or almost surely not happen. It is sometimes known as the Hewitt …   Wikipedia

  • Transformation problem — In 20th century discussions of Karl Marx s economics the transformation problem is the problem of finding a general rule to transform the values of commodities (based on labour according to his labour theory of value) into the competitive prices… …   Wikipedia

  • Bruno de Finetti — Born 13 June 1906(1906 06 13) Innsbruck, Austria …   Wikipedia

  • Sample maximum and minimum — Box plots of the Michelson–Morley experiment, showing sample maximums and minimums. In statistics, the maximum and sample minimum, also called the largest observation, and smallest observation, are the values of the greatest and least elements of …   Wikipedia