Examples of "hypergeometric_distribution"
Fisher's noncentral hypergeometric distribution was first given the name extended hypergeometric distribution (Harkness, 1965), and some authors still use this name today.
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.
The following conditions characterize the hypergeometric distribution:
where formula_21 is the (univariate, central) hypergeometric distribution probability.
where formula_35 is the (univariate, central) hypergeometric distribution probability.
The generalisation of this formula is called the hypergeometric distribution.
See hypergeometric distribution and inferential statistics for further on the distribution of hits.
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 distribution of the balls that are not drawn is a complementary Wallenius' noncentral hypergeometric distribution.
This scenario will give a distribution of the types of fish caught that is equal to Wallenius’ noncentral hypergeometric distribution.
A random variable formula_19 follows the hypergeometric distribution if its probability mass function (pmf) is given by
Fisher showed that the probability of obtaining any such set of values was given by the hypergeometric distribution:
The two distributions are both equal to the (central) hypergeometric distribution when the odds ratio is 1.
The two distributions are both equal to the (central) hypergeometric distribution when the odds ratio is 1.
Negative-hypergeometric distribution (like the hypergeometric distribution) deals with draws "without replacement", so that the probability of success is different in each draw. In contrast, negative-binomial distribution (like the binomial distribution) deals with draws "with replacement", so that the probability of success is the same and the trials are independent. The following table summarizes the four distributions related to drawing items:
Fisher's noncentral hypergeometric distribution is useful for models of biased sampling or biased selection where the individual items are sampled independently of each other with no competition. The bias or odds can be estimated from an experimental value of the mean. Use Wallenius' noncentral hypergeometric distribution instead if items are sampled one by one with competition.
Fisher's noncentral hypergeometric distribution has previously been given the name "extended hypergeometric distribution", but this name is rarely used in the scientific literature, except in handbooks that need to distinguish between the two distributions. Some scientists are strongly opposed to using this name.
The balls that are "not" taken in the urn experiment have a distribution that is different from Wallenius' noncentral hypergeometric distribution, due to a lack of symmetry. The distribution of the balls not taken can be called the complementary Wallenius' noncentral hypergeometric distribution.
In probability theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities are modified by weight factors. It can also be defined as the conditional distribution of two or more binomially distributed variables dependent upon their fixed sum.
If a random variable has a hypergeometric distribution with population size , number of success states } in the population, and draws }, then the factorial moments of are