Wallenius' Noncentral Hypergeometric Distribution

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In probability theory and statistics, Wallenius' noncentral hypergeometric distribution 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 ¿1 and each white ball has the weight ¿2. We will say that the odds ratio is ¿ = ¿1 / ¿2. 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.