Multivariate discrete distributions | Discrete distributions
Wallenius' noncentral hypergeometric distribution
In probability theory and statistics, Wallenius' noncentral hypergeometric distribution (named after Kenneth Ted Wallenius) is a generalization of the hypergeometric distribution where items are sampled with bias. This distribution can be illustrated as an urn model with bias. Assume, for example, that an urn contains m1 red balls and m2 white balls, totalling N = m1 + m2 balls. Each red ball has the weight ω1 and each white ball has the weight ω2. We will say that the odds ratio is ω = ω1 / ω2. Now we are taking n balls, one by one, in such a way that the probability of taking a particular ball at a particular draw is equal to its proportion of the total weight of all balls that lie in the urn at that moment. The number of red balls x1 that we get in this experiment is a random variable with Wallenius' noncentral hypergeometric distribution. The matter is complicated by the fact that there is more than one noncentral hypergeometric distribution. Wallenius' noncentral hypergeometric distribution is obtained if balls are sampled one by one in such a way that there is competition between the balls. Fisher's noncentral hypergeometric distribution is obtained if the balls are sampled simultaneously or independently of each other. Unfortunately, both distributions are known in the literature as "the" noncentral hypergeometric distribution. It is important to be specific about which distribution is meant when using this name. The two distributions are both equal to the (central) hypergeometric distribution when the odds ratio is 1. The difference between these two probability distributions is subtle. See the Wikipedia entry on noncentral hypergeometric distributions for a more detailed explanation. (Wikipedia).
