Beta prime distribution

Beta Prime

Probability density function
Cumulative distribution function
Parameters	$\alpha >0$ shape (real) $\beta >0$ shape (real)
Support	$x>0\!$
PDF	$f(x)={\frac {x^{{\alpha -1}}(1+x)^{{-\alpha -\beta }}}{B(\alpha ,\beta )}}\!$
CDF	$I_{{{\frac {x}{1+x}}(\alpha ,\beta )}}$ where $I_x(\alpha,\beta)$ is the incomplete beta function
Mean	${\frac {\alpha }{\beta -1}}{\text{ if }}\beta >1$
Mode	${\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!$
Variance	${\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2$
Skewness	${\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {{\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}}{\text{ if }}\beta >3$

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 $x>0$ 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 a Beta function.

The cumulative distribution function is

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

where I is the regularized incomplete beta function.

The expectation value, variance, and other details of the distribution are given in the sidebox; for $\beta >4$, 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.^[1]

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 $-\alpha <k<\beta$, 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 .

Differential equation

$\left\{\left(x^{2}+x\right)f'(x)+f(x)(-\alpha +\beta x+x+1)=0,f(1)={\frac {2^{{-\alpha -\beta }}}{B(\alpha ,\beta )}}\right\}$

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

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

and mode

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

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:

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

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 $kX\sim \beta ^{{'}}(\alpha ,\beta ,p,kq)\,$.
• $\beta ^{{'}}(\alpha ,\beta ,1,1)=\beta ^{{'}}(\alpha ,\beta )\,$

Related distributions and properties

• If ${\displaystyle X\sim F(2\alpha ,2\beta )\,}$ then ${\displaystyle X\sim \beta ^{'}(\alpha ,\beta \,,\,1\,,\,{\tfrac {\alpha }{\beta }})\,}$, or equivalently, ${\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta ^{'}(\alpha ,\beta )}$
• 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 ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,}$ are independent, then ${\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta ^{'}(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}$
• Parametrization 2: If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}$ are independent, then ${\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta ^{'}(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}$
• $\beta ^{{'}}(p,1,a,b)={\textrm {Dagum}}(p,a,b)\,$ 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
• Beta prime distribution is a special case of the type 6 Pearson distribution
• Pareto distribution type II is related to Beta prime distribution
• Pareto distribution type IV is related to Beta prime distribution
• inverted Dirichlet distribution, a generalization of the beta prime distribution

Notes

1. Johnson et al (1995), p248
2. Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934.

References

• Jonhnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
• MathWorld article