In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable

equal to the number of failures needed to get

successes in a sequence of independent Bernoulli trials.

of success on each trial stays constant within any given experiment but varies across different experiments following a beta distribution.

the densities of the negative binomial and beta distributions respectively, we obtain the PMF

of the BNB distribution by marginalization: Noting that the integral evaluates to: we can arrive at the following formulas by relatively simple manipulations.

is an integer, then the PMF can be written in terms of the beta function,: More generally, the PMF can be written or Using the properties of the Beta function, the PMF with integer

can be rewritten as: More generally, the PMF can be written as The PMF is often also presented in terms of the Pochammer symbol for integer

The k-th factorial moment of a beta negative binomial random variable X is defined for

and in this case is equal to The beta negative binomial is non-identifiable which can be seen easily by simply swapping

in the above density or characteristic function and noting that it is unchanged.

The beta negative binomial distribution contains the beta geometric distribution as a special case when either

It can therefore approximate the geometric distribution arbitrarily well.

It also approximates the negative binomial distribution arbitrary well for large

It can therefore approximate the Poisson distribution arbitrarily well for large

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

which implies that the beta negative binomial distribution is heavy tailed and that moments less than or equal to

The beta geometric distribution is an important special case of the beta negative binomial distribution occurring for

In this case the pmf simplifies to This distribution is used in some Buy Till you Die (BTYD) models.

the beta geometric reduces to the Yule–Simon distribution.

However, it is more common to define the Yule-Simon distribution in terms of a shifted version of the beta geometric.

are positive integers, the Beta negative binomial can also be motivated by an urn model - or more specifically a basic Pólya urn model.

red balls (the stopping color) and

At each step of the model, a ball is drawn at random from the urn and replaced, along with one additional ball of the same color.

red colored balls are drawn.

of observed draws of blue balls are distributed according to a

Note, at the end of the experiment, the urn always contains the fixed number

of red balls while containing the random number

can be equivalently generated with the urn initially containing

red balls (the stopping color) and