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	${\displaystyle {\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-it\right)}$

In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind^[1]) 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.