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 defined for with two parameters α and β, having the probability density function:

Beta prime
Probability density function
Cumulative distribution function
Parameters shape (real)
shape (real)
CDF where is the incomplete beta function

where B is the Beta function.

The cumulative distribution function is

where I is the regularized incomplete beta function.

The expectation 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 .


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

Compound gamma distribution

The compound gamma distribution[2] 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(x;a,b) 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.


  • If then .
  • If then .
  • If has an F-distribution, then , or equivalently, .
  • If then .
  • If and are independent, then .
  • Parametrization 1: If are independent, then .
  • Parametrization 2: If are independent, then .
  • 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.


  1. Johnson et al (1995), p 248
  2. Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934.


  • Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
  • MathWorld article
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