# Beta negative binomial distribution

Parameters |
shape (real) shape (real) — number of failures until the experiment is stopped (integer but can be extended to real) |
---|---|

Support |
k ∈ { 0, 1, 2, 3, ... } |

pmf | |

Mean | |

Variance | |

Skewness | |

MGF | undefined |

CF |
where B is the beta function and _{2}F_{1} 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.

## 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

## 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, 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

- Interactive graphic: Univariate Distribution Relationships