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	${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}\!}$
CDF	${\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}}$ where ${\displaystyle I_{x}(\alpha ,\beta )}$ is the regularized incomplete beta function
Mean	${\displaystyle {\frac {\alpha }{\beta -1}}}$ if ${\displaystyle \beta >1}$
Mode	${\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}$
Variance	${\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}$ if ${\displaystyle \beta >2}$
Skewness	${\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}}$ if ${\displaystyle \beta >3}$
Excess kurtosis	${\displaystyle 6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}}$ if ${\displaystyle \beta >4}$
Entropy	${\displaystyle {\begin{aligned}&\log \left(\mathrm {B} (\alpha ,\beta )\right)+(\alpha -1)(\psi (\beta )-\psi (\alpha ))\\+&(\alpha +\beta )\left(\psi (1-\alpha -\beta )-\psi (1-\beta )+{\frac {\pi \sin(\alpha \pi )}{\sin(\beta \pi )\sin((\alpha +\beta )\pi ))}}\right)\end{aligned}}}$ where ${\displaystyle \psi }$ is the digamma function.
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) 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.