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	$\frac{\alpha-1}{\beta+1} \text{ if } \alpha\ge 1\text{, 0 otherwise}\!$
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

$\frac{x^\alpha \cdot _2F_1(\alpha, \alpha+\beta, \alpha+1, -x)}{\alpha \cdot B(\alpha,\beta)}\!$

where $_2F_1$ 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

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

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

• If $X \sim F(\alpha,\beta)\,$ then $\tfrac{\alpha}{\beta} X \sim \beta^{'}(\tfrac{\alpha}{2},\tfrac{\beta}{2})\,$
• If $X \sim \textrm{Beta}(\alpha,\beta)\,$ then $\frac{X}{1-X} \sim \beta^{'}(\alpha,\beta)\,$
• If $X \sim \Gamma(\alpha,1)\,$ and $Y \sim \Gamma(\beta,1)\,$, then $\frac{X}{Y} \sim \beta^{'}(\alpha,\beta)$.
• $\beta^{'}(p,1,a,b) = \textrm{Dagum}(p,a,b)\,$ 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