Beta prime
Probability density function
Beta prime pdf.svg
Cumulative distribution function
Beta prime cdf.svg
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.

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

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.

Generalization

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

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 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.

Properties

Related distributions


Notes

  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.

References