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Different texts (and even different parts of this article) adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at k = 0 or at k = r, whether p denotes the probability of a success or of a failure, and whether r represents success or failure, [1] so identifying the specific parametrization used is crucial in any ...
The first distribution generates zeros. The second distribution, which may be a Poisson distribution, a negative binomial distribution or other count distribution, generates counts, some of which may be zeros. [7] In the statistical literature, different authors may use different names to distinguish zeros from the two distributions.
When one or more parameter(s) of a distribution are random variables, the compound distribution is the marginal distribution of the variable. Examples: If X | N is a binomial (N,p) random variable, where parameter N is a random variable with negative-binomial (m, r) distribution, then X is distributed as a negative-binomial (m, r/(p + qr)).
Related to this distribution are a number of other distributions: the displaced Poisson, the hyper-Poisson, the general Poisson binomial and the Poisson type distributions. The Conway–Maxwell–Poisson distribution, a two-parameter extension of the Poisson distribution with an adjustable rate of decay.
The traditional negative binomial regression model is based on the Poisson-gamma mixture distribution. This model is popular because it models the Poisson heterogeneity with a gamma distribution. Poisson regression models are generalized linear models with the logarithm as the (canonical) link function, and the Poisson distribution function as ...
If these conditions are true, then k is a Poisson random variable; the distribution of k is a Poisson distribution. The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals λ divided by the number of trials, as the number of trials approaches infinity (see Related ...
In the case of count data, a Poisson mixture model like the negative binomial distribution can be proposed instead, in which the mean of the Poisson distribution can itself be thought of as a random variable drawn – in this case – from the gamma distribution thereby introducing an additional free parameter (note the resulting negative ...
Therefore, the Poisson distribution with parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is sufficiently large and p is sufficiently small. According to rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05 [ 36 ] such that np ≤ 1 , or if n > 50 and p < 0.1 such that np < 5 , [ 37 ...