Beta prime distribution


In probability theory and statistics, the beta prime distribution is an absolutely continuous probability distribution 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.
The expected value, variance, and other details of the distribution are given in the sidebox; for, the excess kurtosis is
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.
The mode of a variate X distributed as is.
Its mean is if and its variance is if.
For, the k-th moment is given by
For with this simplifies to
The cdf can also be written as
where is the Gauss's hypergeometric function 2F1 .

Generalization

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

Compound gamma distribution

The compound gamma distribution 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 G is the gamma distribution with shape a and inverse scale b. This relationship can be used to generate random variables with a compound gamma, or beta prime distribution.
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.

Properties