# Beta prime distribution

Parameters Probability density function Cumulative distribution function ${\displaystyle \alpha >0}$ shape (real)${\displaystyle \beta >0}$ shape (real) ${\displaystyle x\in (0,\infty )\!}$ ${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}\!}$ ${\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}}$ where ${\displaystyle I_{x}(\alpha ,\beta )}$ is the incomplete beta function ${\displaystyle {\frac {\alpha }{\beta -1}}{\text{ if }}\beta >1}$ ${\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}$ ${\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2}$ ${\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}{\text{ if }}\beta >3}$ ${\displaystyle {\frac {e^{-t}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-t\right)}$

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 defined for ${\displaystyle x>0}$ with two parameters α and β, having the probability density function:

${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}}$

where B is the Beta function.

${\displaystyle F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\beta \right),}$

where I is the regularized incomplete beta function.

The expectation value, variance, and other details of the distribution are given in the sidebox; for ${\displaystyle \beta >4}$, the excess kurtosis is

${\displaystyle \gamma _{2}=6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}.}$

While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds. The distribution is a Pearson type VI distribution.[1]

The mode of a variate X distributed as ${\displaystyle \beta '(\alpha ,\beta )}$ is ${\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}}$. Its mean is ${\displaystyle {\frac {\alpha }{\beta -1}}}$ if ${\displaystyle \beta >1}$ (if ${\displaystyle \beta \leq 1}$ the mean is infinite, in other words it has no well defined mean) and its variance is ${\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}$ if ${\displaystyle \beta >2}$.

For ${\displaystyle -\alpha , the k-th moment ${\displaystyle E[X^{k}]}$ is given by

${\displaystyle E[X^{k}]={\frac {B(\alpha +k,\beta -k)}{B(\alpha ,\beta )}}.}$

For ${\displaystyle k\in \mathbb {N} }$ with ${\displaystyle k<\beta ,}$ this simplifies to

${\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}$

The cdf can also be written as

${\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot B(\alpha ,\beta )}}}$

where ${\displaystyle {}_{2}F_{1}}$ is the Gauss's hypergeometric function 2F1 .

## Generalization

Two more parameters can be added to form the generalized beta prime distribution.

• ${\displaystyle p>0}$ shape (real)
• ${\displaystyle q>0}$ scale (real)

having the probability density function:

${\displaystyle f(x;\alpha ,\beta ,p,q)={\frac {p\left({\frac {x}{q}}\right)^{\alpha p-1}\left(1+\left({\frac {x}{q}}\right)^{p}\right)^{-\alpha -\beta }}{qB(\alpha ,\beta )}}}$

with mean

${\displaystyle {\frac {q\Gamma \left(\alpha +{\tfrac {1}{p}}\right)\Gamma (\beta -{\tfrac {1}{p}})}{\Gamma (\alpha )\Gamma (\beta )}}\quad {\text{if }}\beta p>1}$

and mode

${\displaystyle q\left({\frac {\alpha p-1}{\beta p+1}}\right)^{\tfrac {1}{p}}\quad {\text{if }}\alpha p\geq 1}$

Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution

### Compound gamma distribution

The compound gamma distribution[2] is the generalization of the beta prime when the scale parameter, q is added, but where p = 1. It is so named because it is formed by compounding two gamma distributions:

${\displaystyle \beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr}$

where G(x;a,b) is the gamma distribution with shape a and inverse scale b. This relationship can be used to generate random variables with a compound gamma, or beta prime distribution.

The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by q and the variance by q2.

## Properties

• If ${\displaystyle X\sim \beta '(\alpha ,\beta )}$ then ${\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )}$.
• If ${\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}$ then ${\displaystyle kX\sim \beta '(\alpha ,\beta ,p,kq)}$.
• ${\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )}$

## Related distributions and properties

• If ${\displaystyle X\sim F(2\alpha ,2\beta )}$ has an F-distribution, then ${\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )}$, or equivalently, ${\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})}$.
• If ${\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )}$ then ${\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}$.
• If ${\displaystyle X\sim \Gamma (\alpha ,1)}$ and ${\displaystyle Y\sim \Gamma (\beta ,1)}$ are independent, then ${\displaystyle {\frac {X}{Y}}\sim \beta '(\alpha ,\beta )}$.
• Parametrization 1: If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})}$ are independent, then ${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}$.
• Parametrization 2: If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})}$ are independent, then ${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}$.
• ${\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)}$ the Dagum distribution
• ${\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)}$ the Singh–Maddala distribution.
• ${\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )}$ the log logistic distribution.
• The beta prime distribution is a special case of the type 6 Pearson distribution.
• If X has a Pareto distribution with minimum ${\displaystyle x_{m}}$ and shape parameter ${\displaystyle \alpha }$, then ${\displaystyle X-x_{m}\sim \beta ^{\prime }(1,\alpha )}$.
• If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter ${\displaystyle \alpha }$ and scale parameter ${\displaystyle \lambda }$, then ${\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}$.
• If X has a standard Pareto Type IV distribution with shape parameter ${\displaystyle \alpha }$ and inequality parameter ${\displaystyle \gamma }$, then ${\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )}$, or equivalently, ${\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)}$.
• The inverted Dirichlet distribution is a generalization of the beta prime distribution.

## Notes

1. ^ a b Johnson et al (1995), p 248
2. ^ Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934.