In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, a beta negative binomial distribution is the probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 of a discrete random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

 X equal to the number of failures in needed to get n successes in a sequence of independent Bernoulli trial

Bernoulli trial

In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure"....

s 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

Compound probability distribution

In probability theory, a compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution F with an unknown parameter θ that is distributed according to some other distribution G, and then...

.

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

In the above, NB(n, p) is the negative binomial distribution

Negative binomial distribution

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number of failures occur...

 and B(α, β) is the beta distribution.