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
has a beta distribution, then the odds
has a beta prime distribution.
Quick Facts Parameters, Support ...
Beta prime
Probability density function |
Cumulative distribution function |
Parameters |
shape (real)
shape (real) |
---|
Support |
![{\displaystyle x\in [0,\infty )\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/be00d03e32dc2701e685f3f746dee4702dfdd208) |
---|
PDF |
![{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/1b14e2ca05e0237f7b98dca313f3c3f8658fe1a4) |
---|
CDF |
where is the incomplete beta function |
---|
Mean |
![{\displaystyle {\frac {\alpha }{\beta -1}}{\text{ if }}\beta >1}](//wikimedia.org/api/rest_v1/media/math/render/svg/3e26f4eb2c328cfddbaae3bac89e74e596b29c2a) |
---|
Mode |
![{\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/6317382f881af2a06b1acd41dbccc9f6effcc97d) |
---|
Variance |
![{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2}](//wikimedia.org/api/rest_v1/media/math/render/svg/5b295ffdae2069977c7e9b4d1347e331314e22d0) |
---|
Skewness |
![{\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}{\text{ if }}\beta >3}](//wikimedia.org/api/rest_v1/media/math/render/svg/2e81a86b913ff240a54f87ecf4c01bb633e994bc) |
---|
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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/d02d27cf5d3f60e5dafdfa9384560412c29bc955) |
---|
Close