# Beta negative binomial distribution

Parameters Probability mass function File:No image available Cumulative distribution function File:No image available $\alpha > 0$ shape (real) $\beta > 0$ shape (real) n ∈ N0 — number of trials k ∈ { 0, 1, 2, 3, ... } $\frac{n^{(k)}\alpha^{(n)}\beta^{(k)}}{k!(\alpha+\beta)^{(n)}(n+\alpha+\beta)^{(k)}}$ Where $x^{(n)}$ is the rising Pochhammer symbol $\begin{cases} \frac{n\beta}{\alpha-1} & \text{if}\ \alpha>1 \\ \infty & \text{otherwise}\ \end{cases}$ $\begin{cases} \frac{n(\alpha+n-1)\beta(\alpha+\beta-1)}{(\alpha-2){(\alpha-1)}^2} & \text{if}\ \alpha>2 \\ \infty & \text{otherwise}\ \end{cases}$ $\begin{cases} \frac{(\alpha+2n-1)(\alpha+2\beta-1)}{(\alpha-3)\sqrt{\frac{n(\alpha+n-1)\beta(\alpha+\beta-1)}{\alpha-2}}} & \text{if}\ \alpha>3 \\ \infty & \text{otherwise}\ \end{cases}$

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 n 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

$X \mid p \sim \mathrm{NB}(n,p),$

where

$p \sim \textrm{B}(\alpha,\beta),$

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

$X \sim \mathrm{BNB}(n,\alpha,\beta).$

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

Recurrence relation

$\left\{(k+1) p(k+1) (\alpha +\beta +k+n)+(\beta +k) (-k-n) p(k)=0,p(0)=\frac{(\alpha )_n}{(\alpha +\beta )_n}\right\}$

## PMF expressed with Gamma

Since the rising Pochhammer symbol can be expressed in terms of the Gamma function, the numerator of the PMF as given can be expressed as:

$\frac{\Gamma(n+k)\Gamma(\alpha+n)\Gamma(\beta+k)}{\Gamma(n)\Gamma(\alpha)\Gamma(\beta)}$.

Likewise, the denominator can be rewritten as:

$\frac{\Gamma(\alpha+\beta)\Gamma(\alpha+\beta+n)}{k!\Gamma(\alpha+\beta+n)\Gamma(\alpha+\beta+n+k)}$,

and the two ${\Gamma(\alpha+\beta+n)}$ terms cancel out, leaving:

$\frac{\Gamma(\alpha+\beta)}{k!\Gamma(\alpha+\beta+n+k)}$.

As $\frac{\Gamma(n+k)}{k!\Gamma(n)} = \binom{n+k-1}k$, the PMF can be rewritten as:

$\binom{n+k-1}k\frac{\Gamma(\alpha+n)\Gamma(\beta+k)\Gamma(\alpha+\beta)}{\Gamma(\alpha+\beta+n+k)\Gamma(\alpha)\Gamma(\beta)}$.

### PMF expressed with Beta

Using the beta function, the PMF is:

$\binom{n+k-1}k\frac{\Beta(\alpha+n,\beta+k)}{\Beta(\alpha,\beta)}$.

Replacing the binomial coefficient by a beta function, the PMF can also be written:

$\frac{\Beta(\alpha+n,\beta+k)}{k\Beta(\alpha,\beta)\Beta(n,k)}$.

## Notes

1. ^ a b 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, 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