Exchangeable random variables: Difference between revisions

Revision as of 19:47, 2 September 2021

Revision as of 14:59, 24 April 2023

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* Any [[convex combination]] or [[mixture distribution]] of [[iid]] sequences of random variables is exchangeable. A converse proposition is [[de Finetti's theorem]].<ref>Spizzichino, Fabio ''Subjective probability models for lifetimes''. Monographs on Statistics and Applied Probability, 91. ''Chapman & Hall/CRC'', Boca Raton, FL, 2001. xx+248 pp.&nbsp;{{ISBN|1-58488-060-0}}

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* Suppose an [[urn model|urn]] contains ''<math>n''</math> red and ''<math>m''</math> blue marbles. Suppose marbles are drawn without replacement until the urn is empty. Let ''X''<submath>''i''X_i</submath> be the indicator random variable of the event that the ''<math>i''</math>-th marble drawn is red. Then {''X''<sub>''i''</submath>\left\{ X_i \right\}<sub>''_{i''=1,...'' \dots, n+m''}</submath> is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.

* Suppose an urn contains <math>n</math> red and <math>m</math> blue marbles. Further suppose a marble is drawn from the urn and then replaced, with an extra marble of the same colour. Let <math>X_i</math> be the indicator random variable of the event that the <math>i</math>-th marble drawn is red. Then <math>\left\{ X_i \right\}_{i\in \N}</math> is an exchangeable sequence. This model is called [[Polya's urn]].

* Let <math>(X, Y)</math> have a [[bivariate normal distribution]] with parameters <math>\mu = 0</math>, <math>\sigma_x = \sigma_y = 1</math> and an arbitrary [[Pearson product-moment correlation coefficient|correlation coefficient]] <math>\rho\in (-1, 1)</math>. The random variables <math>X</math> and <math>Y</math> are then exchangeable, but independent only if <math>\rho=0</math>. The [[density function]] is <math>p(x, y) = p(y, x) \propto \exp\left[-\frac{1}{2(1-\rho^2)}(x^2+y^2-2\rho xy)\right].</math>