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Beta Prime Distribution

A distribution with probability function

P(x)=(x^(alpha-1)(1+x)^(-alpha-beta))/(B(alpha,beta)),

where B is a beta function. The mode of a variate distributed as beta^'(alpha,beta) is

x^^=(alpha-1)/(beta+1).

If X is a beta^'(alpha,beta) variate, then 1/X is a beta^'(beta,alpha) variate. If X is a beta(alpha,beta) variate, then (1-X)/X and X/(1-X) are beta^'(beta,alpha) and beta^'(alpha,beta) variates. If X and Y are gamma(alpha_1) and gamma(alpha_2) variates, then X/Y is a beta^'(alpha_1,alpha_2) variate. If X^2/2 and Y^2/2 are gamma(1/2) variates, then Z^2=(X/Y)^2 is a beta^'(1/2,1/2) variate.

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Cite this as:

Weisstein, Eric W. "Beta Prime Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BetaPrimeDistribution.html

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