In our present context, the hypergeometric is, in effect, a binomial distribution
with conditional probabilities, indicated by the convention of placing the power n in parentheses, as the power (4) in expansion (2).
One approach to deriving the negative binomial distribution
is to assume that the count for each sampling unit is distributed as a Poisson variable with mean [[Mu].
Fishman  describes a method for sampling from the binomial distribution
by using the acceptance/rejection technique with a uniform majorizing function, and later improves the idea by changing to a Poisson majorizing function .
They therefore use the overdispersed negative binomial distribution
and anticipate that the negative binomial provides a much better fit in the tail region (see Table 1), an observation confirmed by the Pearson [chi square] test using the parameters based on MLE.
Tests of significance were based on the [chi square] statistic for the binomial distribution
of the proportion of positive tows (McCullagh and Nelder, 1989).
The k parameter of the negative binomial distribution
estimated by the maximum likelihood method is calculated iteratively and is the value that equates the two members of the following (Bliss and Fisher, 1953):
t] can then actually be obtained by using the relationship between the cumulative beta distribution and the cumulative binomial distribution
function as follows (Daly  and Johnson et al.
We fit the transmission data from patients within subgroups to the negative binomial distribution
with mean R and dispersion parameter k, which characterizes individual variation in transmission, including the likelihood of superspreading events (i.
The parameters a and b of the beta-binomial model can be chosen to provide flexibility to handle many possible situations in health services research that have this "probability" nature of constraining between 0 and 1, and are more diffuse than the over-dispersion capabilities of the negative binomial distribution
(Morris and Lock 2009).
Negative binomial distributions
were fitted to all pairs of players within a side so that interactions between players could be simulated prior to a match.
The three most common statistical distributions are the normal, Poisson and binomial distributions
are used to model situations where there are two outcomes, such as pass or fail.