Beta prime
Probability density function
Cumulative distribution function
Parameters shape (real)
shape (real)
CDF where is the incomplete beta function
MGF 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[1]) 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.[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 .

Alternative parameterization

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 :

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 .

Compound gamma distribution

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.)


Related distributions


  1. ^ a b Johnson et al (1995), p 248
  2. ^ 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.
  3. ^ Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934. S2CID 123366328.