Beta prime distribution

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. If has a beta distribution, then the odds has a beta prime distribution.

Beta prime

Probability density function
Cumulative distribution function
Parameters	shape (real) shape (real)
Support
PDF
CDF	where is the incomplete beta function
Mean
Mode
Variance
Skewness
MGF	Does not exist
CF

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.^[1]

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 (^[2] 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:

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. If   and   for  , then  .

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   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.)

• If   then  .
• If  , and  , then  .
• If   then  .
•  
• If   and   two iid variables, then   with   and  , as the beta prime distribution is infinitely divisible.
• More generally, let   iid variables following the same beta prime distribution, i.e.  , then the sum   with   and  .

• If   has an F-distribution, then  , or equivalently,  .
• If   then  .
• If   then  .
• For gamma distribution parametrization I:
  • If   are independent, then  . Note   are all scale parameters for their respective distributions.
• For gamma distribution parametrization II:
  • If   are independent, then  . The   are rate parameters, while   is a scale parameter.
  • If   and  , then  . The   are rate parameters for the gamma distributions, but   is the scale parameter for the beta prime.
•   the Dagum distribution
•   the Singh–Maddala distribution.
•   the log logistic distribution.
• The beta prime distribution is a special case of the type 6 Pearson distribution.
• If X has a Pareto distribution with minimum   and shape parameter  , then  .
• If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter   and scale parameter  , then  .
• If X has a standard Pareto Type IV distribution with shape parameter   and inequality parameter  , then  , or equivalently,  .
• The inverted Dirichlet distribution is a generalization of the beta prime distribution.
• If  , then   has a generalized logistic distribution. More generally, if  , then   has a scaled and shifted generalized logistic distribution.

• Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
• Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021), "A new regression model for positive random variables with skewed and long tail", Metron, 79: 33–55, doi:10.1007/s40300-021-00203-y, S2CID 233534544

• MathWorld article