Probability density function
Cumulative distribution function
where is the incomplete beta function|
Does not exist|
In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind) is an absolutely continuous probability distribution. If 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.
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 (if the mean is infinite, in other words it has no well defined mean) 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 .
The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters ( p. 36).
Consider the parameterization μ = α/(β-1) and ν = β- 2, i.e., α = μ( 1 + ν) and
β = 2 + ν. Under this parameterization
E[Y] = μ and Var[Y] = μ(1 + μ)/ν.
Two more parameters can be added to form the generalized beta prime distribution :
- shape (real)
- scale (real)
having the probability density function:
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. If and for , then .
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 is the gamma pdf with shape and inverse scale .
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.
Another way to express the compounding is if and , then . (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.)