Beta negative binomial distribution

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Beta Negative Binomial
Parameters shape (real)
shape (real)
— number of failures until the experiment is stopped (integer but can be extended to real)
Supportk ∈ { 0, 1, 2, 3, ... }
CF where B is the beta function and 2F1 is the hypergeometric function.

In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable X equal to the number of failures needed to get r successes in a sequence of independent Bernoulli trials where the probability p of success on each trial is constant within any given experiment but is itself a random variable following a beta distribution, varying between different experiments. Thus the distribution is a compound probability distribution.

This distribution has also been called both the inverse Markov-Pólya distribution and the generalized Waring distribution.[1] A shifted form of the distribution has been called the beta-Pascal distribution.[1]

If parameters of the beta distribution are α and β, and if


then the marginal distribution of X is a beta negative binomial distribution:

In the above, NB(rp) is the negative binomial distribution and B(αβ) is the beta distribution.


If is an integer, then the PMF can be written in terms of the beta function,:


More generally the PMF can be written


PMF expressed with Gamma[edit]

Using the properties of the Beta function, the PMF with integer can be rewritten as:


More generally, the PMF can be written as


PMF expressed with the rising Pochammer symbol[edit]

The PMF is often also presented in terms of the Pochammer symbol for integer


The beta negative binomial distribution contains the beta geometric distribution as a special case when . It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large and . It can therefore approximate the Poisson distribution arbitrarily well for large , and .

By Stirling's approximation to the beta function, it can be easily shown that

which implies that the beta negative binomial distribution is heavy tailed.


  1. ^ a b Johnson et al. (1993)


  • Jonhnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley ISBN 0-471-54897-9 (Section 6.2.3)
  • Kemp, C.D.; Kemp, A.W. (1956) "Generalized hypergeometric distributions, Journal of the Royal Statistical Society, Series B, 18, 202–211
  • Wang, Zhaoliang (2011) "One mixed negative binomial distribution with application", Journal of Statistical Planning and Inference, 141 (3), 1153-1160 doi:10.1016/j.jspi.2010.09.020

External links[edit]