In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind ) is an absolutely continuous probability distribution. If p\in[0,1] has a beta distribution, then the odds has a beta prime distribution.

Definitions

Beta prime distribution is defined for x > 0 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. 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. The mode of a variate X distributed as is. Its mean is if \beta>1 (if the mean is infinite, in other words it has no well defined mean) and its variance is if \beta>2. For, the k-th moment E[X^k] is given by For with k <\beta, this simplifies to The cdf can also be written as where {}_2F_1 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 ( 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. This generalization can be obtained via the following invertible transformation. If and x=qy^{1/p} for q,p>0, then.

Compound gamma distribution

The compound gamma distribution 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 pdf with shape a and inverse scale b. 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.

Properties

Related distributions