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) 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. 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 . 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. 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 . The compound gamma distribution is the generalization of the beta prime when the scale parameter, q is added, but where p = 1.