Exchangeable random variables

Let $\xi =\{E_{1},\ldots ,E_{n}\}$ be a family of random events, and let $X_{1},\ldots ,X_{n}$ be the indicator function for each of the events in $\xi$. Then $\xi$ is said to be exchangeable if, for any permutation $j_{1},\ldots ,j_{n}$ of the indexes $1,\ldots ,n$, the two random vectors $(X_{1},\ldots ,X_{n})$ and $(X_{j_{1}},\ldots ,X_{j_{n}})$ have the same joint distribution.

With a more general view, a family of generic random variables $(X_{1},\ldots ,X_{n})$ is exchangeable if, for any permutation $j_{1}\ldots ,j_{n}$ of the indexes $1,\ldots ,n$, they have the same joint distribution.

Independent and identically random variables are exchangeable.

An interesting property of exchangeability is that the distribution function $F_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})$ is symmetric in its arguments $(x_{1},\ldots ,x_{n})$.

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