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 ${\frac {\alpha }{\beta -1)){\text{ if ))\beta >1$ ${\frac {\alpha -1}{\beta +1)){\text{ if ))\alpha \geq 1{\text{, 0 otherwise))\!$ ${\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2))}{\text{ if ))\beta >2$ ${\frac {2(2\alpha +\beta -1)}{\beta -3)){\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)))}{\text{ if ))\beta >3$ Does not exist ${\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )))G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix))\;\right|\,-it\right)$ 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.

## Definitions

Beta prime distribution is 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 .

### Alternative parameterization

The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters ( p. 36).

Consider the parameterization μ = α/(β-1) and ν = β- 2, i.e., α = μ( 1 + ν) and β = 2 + ν. Under this parameterization E[Y] = μ and Var[Y] = μ(1 + μ)/ν.

### Generalization

Two more parameters can be added to form the generalized beta prime distribution $\beta '(\alpha ,\beta ,p,q)$ :

• $p>0$ 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

• 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 $X\sim \Gamma (\alpha ,\theta )$ and $Y\sim \Gamma (\beta ,\theta )$ 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 ${\dfrac {X}{x_{m))}-1\sim \beta ^{\prime }(1,\alpha )$ .
• 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.

1. ^ a b Johnson et al (1995), p 248
2. ^ Bourguignon, M.; Santos-Neto,M.; de Castro,M. (2021). "A new regression model for positive random variables with skewed and long tail". Metron. 79: 33–55. doi:10.1007/s40300-021-00203-y. S2CID 233534544.
3. ^ Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934. S2CID 123366328.
• Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
• Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021), "A new regression model for positive random variables with skewed and long tail", Metron, 79: 33–55, doi:10.1007/s40300-021-00203-y, S2CID 233534544