Probability 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 |
 |
---|
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.
Definitions
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
![E[X^{k}]={\frac {B(\alpha +k,\beta -k)}{B(\alpha ,\beta ))).](https://wikimedia.org/api/rest_v1/media/math/render/svg/30c6530cfd83409026129cc40968169281f41081)
For
with
this simplifies to
![{\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f1689a0ef95460a83f9f53462da32a9b1e8f04)
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.
Generalization
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
.
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.)