Content deleted Content added
→Covariance and Correlation: ce, including making italics legible |
→Examples: making italics legible |
||
Line 69:
* 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. {{ISBN|1-58488-060-0}}
</ref>
* Suppose an [[urn model|urn]] contains ''n'' red and ''m'' blue marbles. Suppose marbles are drawn without replacement until the urn is empty. Let ''X''<sub>''i''</sub> be the indicator random variable of the event that the ''i''-th marble drawn is red. Then {''X''<sub>''i''</sub>}<sub>''i''=1,...''n''</sub> is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.
* 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>
| |||