Wallenius' Noncentral Hypergeometric Distribution
Wallenius' distribution can be explained as follows. Assume that an urn contains red balls and white balls, totalling balls. balls are drawn at random from the urn one by one without replacement. Each red ball has the weight, and each white ball has the weight . We assume that the probability of taking a particular ball is proportional to its weight. The physical property that determines the odds may be something else than weight, such as size or slipperiness or whatever, but it is convenient to use the word weight for the odds parameter.
The probability that the first ball picked is red is equal to the weight fraction of red balls:
The probability that the second ball picked is red depends on whether the first ball was red or white. If the first ball was red then the above formula is used with reduced by one. If the first ball was white then the above formula is used with reduced by one.
The important fact that distinguishes Wallenius' distribution is that there is competition between the balls. The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. And the weight of the competing balls depends on the outcomes of all preceding draws.
A multivariate version of Wallenius' distribution is used if there are more than two different colors.
The distribution of the balls that are not drawn is a complementary Wallenius' noncentral hypergeometric distribution.
Read more about this topic: Noncentral Hypergeometric Distributions
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