The negative binomial distribution

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The negative binomial distribution describes the probability of  successes in a sequence of independent experiments each with likelihood of success of  that arise before there are  failures. In this interpretation  is a positive integer, but the distributional definition can also be extended to real values of . Note: different texts adopt slightly different definitions, e.g. with support starting at  not  and/or with  denoting probability of failure rather than probability of success.


Distribution name

Negative binomial distribution

Common notation


 = number of failures ()

 = probability of success in each experiment ()


Probability mass function

If  is non-integral then is:

Cumulative distribution function




(Excess) kurtosis

Characteristic function

Other comments

The geometric distribution is the same as the negative binomial distribution with parameter . Its pdf and cdf are therefore:


For the special case where  is an integer the negative binomial distribution is also called the Pascal distribution. The Poisson distribution is also a limiting case of the negative binomial:


Nematrian web functions


Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “negative binomial”. For details of other supported probability distributions see here.


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