In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind[1]) is an absolutely continuous probability distribution.

has a beta distribution, then the odds

has a beta prime distribution.

Beta prime distribution is defined for

with two parameters α and β, having the probability density function: where B is the Beta function.

The cumulative distribution function is where I is the regularized incomplete beta function.

While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds.

The distribution is a Pearson type VI distribution.

[1] The mode of a variate X distributed as

the mean is infinite, in other words it has no well defined mean) and its variance is

, the k-th moment

this simplifies to The cdf can also be written as where

is the Gauss's hypergeometric function 2F1 .

The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters ([2] p. 36).

Consider the parameterization μ = α/(β − 1) and ν = β − 2, i.e., α = μ(1 + ν) and β = 2 + ν.

Two more parameters can be added to form the generalized beta prime distribution

: having the probability density function: with mean and mode Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution.

This generalization can be obtained via the following invertible transformation.

The compound gamma distribution[3] is the generalization of the beta prime when the scale parameter, q is added, but where p = 1.

It is so named because it is formed by compounding two gamma distributions: where

is the gamma pdf with shape

The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by q and the variance by q2.

This gives one way to generate random variates with compound gamma, or beta prime distributions.

Another is via the ratio of independent gamma variates, as shown below.