Exchangeable random variables

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

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

Independent and identically random variables are exchangeable.

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

See 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