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Milds # Beta prime distribution

Parameters Probability density function Cumulative distribution function $\alpha >0$ shape (real)$\beta >0$ shape (real) $x\in [0,\infty )\!$ $f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}\!$ $I_{{\frac {x}{1+x}}(\alpha ,\beta )}$ where $I_{x}(\alpha ,\beta )$ is the incomplete beta function ${\frac {\alpha }{\beta -1}}{\text{ if }}\beta >1$ ${\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!$ ${\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2$ ${\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}{\text{ if }}\beta >3$ ${\frac {e^{-t}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-t\right)$ In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind) is an absolutely continuous probability distribution defined for $x>0$ with two parameters α and β, having the probability density function:

$f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}$ where B is the Beta function.

$F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\beta \right),$ where I is the regularized incomplete beta function.

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

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

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

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

$E[X^{k}]={\frac {B(\alpha +k,\beta -k)}{B(\alpha ,\beta )}}.$ For $k\in \mathbb {N}$ with $k<\beta ,$ this simplifies to

$E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.$ The cdf can also be written as

${\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot B(\alpha ,\beta )}}$ where ${}_{2}F_{1}$ is the Gauss's hypergeometric function 2F1 .

## Generalization

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

• $p>0$ shape (real)
• $q>0$ scale (real)

having the probability density function:

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

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

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

$\beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr$ 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 $X\sim \beta '(\alpha ,\beta )$ then ${\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )$ .
• If $X\sim \beta '(\alpha ,\beta ,p,q)$ then $kX\sim \beta '(\alpha ,\beta ,p,kq)$ .
• $\beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )$ • If $X_{1}\sim \beta '(\alpha ,\beta )$ and $X_{2}\sim \beta '(\alpha ,\beta )$ two iid variables, then $Y=X_{1}+X_{2}\sim \beta '(\gamma ,\delta )$ with $\gamma ={\frac {2\alpha (\alpha +\beta ^{2}-2\beta +2\alpha \beta -4\alpha +1)}{(\beta -1)(\alpha +\beta -1)}}$ and $\delta ={\frac {2\alpha +\beta ^{2}-\beta +2\alpha \beta -4\alpha }{\alpha +\beta -1}}$ , as the beta prime distribution is infinitely divisible.
• More generally, let $X_{1},...,X_{n}n$ iid variables following the same beta prime distribution, i.e. $\forall i,1\leq i\leq n,X_{i}\sim \beta '(\alpha ,\beta )$ , then the sum $S=X_{1}+...+X_{n}\sim \beta '(\gamma ,\delta )$ with $\gamma ={\frac {n\alpha (\alpha +\beta ^{2}-2\beta +n\alpha \beta -2n\alpha +1)}{(\beta -1)(\alpha +\beta -1)}}$ and $\delta ={\frac {2\alpha +\beta ^{2}-\beta +n\alpha \beta -2n\alpha }{\alpha +\beta -1}}$ .

## Related distributions and properties

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