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 ${\displaystyle p\in [0,1]}$ has a beta distribution, then the odds ${\displaystyle {\frac {p}{1-p}}}$ has a beta prime distribution.

Beta prime

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \alpha >0}$ shape (real) ${\displaystyle \beta >0}$ shape (real)
Support	${\displaystyle x\in [0,\infty )\!}$
PDF	${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}\!}$
CDF	${\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}}$ where ${\displaystyle I_{x}(\alpha ,\beta )}$ is the incomplete beta function
Mean	${\displaystyle {\frac {\alpha }{\beta -1}}{\text{ if }}\beta >1}$
Mode	${\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}$
Variance	${\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2}$
Skewness	${\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}{\text{ if }}\beta >3}$
MGF	Does not exist
CF	${\displaystyle {\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-it\right)}$