Beta negative binomial distribution

Beta Negative Binomial

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

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. A shifted form of the distribution has been called the beta-Pascal distribution.

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(n, p) is the negative binomial distribution and B(α, β) is the beta distribution.

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:

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

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

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

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