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Negative polynomial distribution

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negative multinomial distribution

The joint probability distribution (cf. also Joint distribution) of random variables that take non-negative integer values defined by the formula

(*)

where and (, ; ) are parameters. A negative multinomial distribution is a multi-dimensional discrete distribution — a distribution of a random vector with non-negative integer components.

The generating function of the negative polynomial distribution with parameters has the form

A negative multinomial distribution arises in the following multinomial scheme. Successive independent trials are carried out, and in each trial different outcomes with labels are possible, having probabilities , respectively. The trials continue up to the -th appearance of the outcome with label 0 (here is an integer). If is the number of appearances of the outcome with label , , during the trials, then formula (*) expresses the probability of the appearance of outcomes with labels , equal, respectively, times, up to the -th appearance of the outcome 0. A negative multinomial distribution in this sense is a generalization of a negative binomial distribution, coinciding with it when .

If a random vector has, conditionally on , a multinomial distribution with parameters , and if the parameter is itself a random variable having a negative binomial distribution with parameters , , then the marginal distribution of the vector , given the condition , is the negative multinomial distribution with parameters , .


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References

[a1] J. Neyman, "Proceedings of the international symposium on discrete distributions" , Montreal (1963)
How to Cite This Entry:
Negative polynomial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_polynomial_distribution&oldid=47953
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article