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The negative binomial distribution NB(r, p) can be represented as a compound Poisson distribution: Let () denote a sequence of independent and identically distributed random variables, each one having the logarithmic series distribution Log(p), with probability mass function
The negative binomial series includes the case of the geometric series, the power series [1] = = (which is the negative binomial series when =, convergent in the disc | | <) and, more generally, series obtained by differentiation of the geometric power series: = ()! with =, a positive integer.
The negative hypergeometric distribution, a distribution which describes the number of attempts needed to get the nth success in a series of Yes/No experiments without replacement. The Poisson binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with different success probabilities.
The beta negative binomial is non-identifiable which can be seen easily by simply swapping and in the above density or characteristic function and noting that it is unchanged. Thus estimation demands that a constraint be placed on r {\displaystyle r} , β {\displaystyle \beta } or both.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x 0, p)) to more than two outcomes. [ 1 ] As with the univariate negative binomial distribution, if the parameter x 0 {\displaystyle x_{0}} is a positive integer, the negative multinomial distribution has an ...
The proof is similar, but uses the binomial series expansion with negative integer exponents. When j = k, equation gives the hockey-stick identity = = (+ ...
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).