Built-in Mathematica Symbol

BetaNegativeBinomialDistribution

,

, n]
represents a beta negative binomial mixture distribution with beta distribution parameters

and

, and

successful trials.

The beta negative binomial distribution is a negative binomial distribution whose probability parameter p follows a beta distribution with shape parameters and . »

BetaNegativeBinomialDistribution allows , and n to be any positive real numbers.

Mean and variance of a beta negative binomial distribution:

Probability density function:

Mean and variance of a beta negative binomial distribution:

Probability density function:

Generate a set of pseudorandom numbers that have a beta negative binomial distribution:

Properties based on higher-order moments:

The probability of at least 50 failures before 10 successes assuming a beta distribution on p:

Plot the cumulative distribution function of a beta negative binomial distribution:

Density functions for beta negative binomial distributions depend heavily on the parameters

and

:

The probability of getting negative integers, integers beyond n, or non-integer numbers is zero:

Beta negative binomial is a mixture of NegativeBinomialDistribution and BetaDistribution:

,

or n is non-positive:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: