# Beta negative binomial distribution

Parameters shape (real) shape (real) — number of failures until the experiment is stopped (integer but can be extended to real) k ∈ { 0, 1, 2, 3, ... } undefined 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

where

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

In the above, NB(r, p) is the negative binomial distribution and B(α, β) is the beta distribution.

Recurrence relation

## Definition

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

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

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

## Properties

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.

## Notes

1. Johnson et al. (1993)

## References

• 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, 202211
• 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