Beta prime distribution

Beta prime distribution

Probability distribution
name =Beta Prime| type =density


parameters =alpha > 0 shape (real)
eta > 0 shape (real)
support =x > 0!
pdf =f(x) = frac{x^{alpha-1} (1+x)^{-alpha -eta
cdf =frac{x^alpha cdot _2F_1(alpha, alpha+eta, alpha+1, -x)}{alpha cdot B(alpha,eta)}!where _2F_1 is the Gauss's hypergeometric function 2F1
mean =frac{alpha}{eta-1}
median =
mode =frac{alpha-1}{eta+1}!
variance =frac{alpha(alpha+eta-1)}{(eta-2)(eta-1)^2}
skewness =
kurtosis =
entropy =
mgf =
char =
A Beta Prime Distribution is a probability distribution defined for x>0 with two parameters (of positive real part), α and β, having the probability density function:

f(x) = frac{x^{alpha-1} (1+x)^{-alpha -eta{B(alpha,eta)}

where B is a Beta function. This distribution is also known [Johnson et al (1995), p248] as the beta distribution of the second kind. It is basically the same as the F distribution--if b is distributed as the beta prime distribution Beta'(α,β), then bβ/α obeys the F distribution with 2α and 2β degrees of freedom. The distribution is a Pearson type VI distribution [Johnson et al (1995), p248] .

The mode of a variate X distributed as eta^{'}(alpha,eta) is hat{X} = frac{alpha-1}{eta+1}.Its mean is frac{alpha}{eta-1} if eta>1 (if eta<=1 the mean is infinite, in other words it has no well defined mean)and its variance isfrac{alpha(alpha+eta-1)}{(eta-2)(eta-1)^2} if eta>2.

If X is a eta^{'}(alpha,eta) variate then frac{1}{X} is a eta^{'}(eta,alpha) variate.

If X is a eta(alpha,eta) then frac{1-X}{X} and frac{X}{1-X} are eta^{'}(eta,alpha) and eta^{'}(alpha,eta) variates.

If X and Y are gamma(alpha_1) and gamma(alpha_2) variates, then frac{X}{Y} is a eta^{'}(alpha_1,alpha_2) variate.



Jonhnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariuate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0

[ MathWorld article]

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