Beta prime distribution
From Wikipedia, the free encyclopedia
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:
<math>f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}</math>
where <math>B</math> is a Beta function. 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 mode of a variate <math>X</math> distributed as <math>\beta^{'}(\alpha,\beta)</math> is <math>\hat{X} = \frac{\alpha-1}{\beta+1}</math>.
If X is a <math>\beta^{'}(\alpha,\beta)</math> variate then <math>\frac{1}{X}</math> is a <math>\beta^{'}(\beta,\alpha)</math> variate.
If X is a <math>\beta^{'}(\alpha,\beta)</math> then <math>\frac{1-X}{X}</math> and <math>\frac{X}{1-X}</math> are <math>\beta^{'}(\beta,\alpha)</math> and <math>\beta^{'}(\alpha,\beta)</math> variates.
If X and Y are <math>\gamma(\alpha_1)</math> and <math>\gamma(\alpha_2)</math> variates, then <math>\frac{X}{Y}</math> is a <math>\beta^{'}(\alpha_1,\alpha_2)</math> variate.
