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 $X\sim F(2\alpha ,2\beta )\,$ then $X\sim \beta ^{'}(\alpha ,\beta \,,\,1\,,\,{\tfrac {\alpha }{\beta }})\,$ , or equivalently, ${\tfrac {\alpha }{\beta }}X\sim \beta ^{'}(\alpha ,\beta )$
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 $X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,$ are independent, then ${\frac {X_{1}}{X_{2}}}\sim \beta ^{'}(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})$
Parametrization 2: If $X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,$ are independent, then ${\frac {X_{1}}{X_{2}}}\sim \beta ^{'}(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})$
$\beta ^{{'}}(p,1,a,b)={\textrm {Dagum}}(p,a,b)\,$ 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 2 Johnson et al (1995), p248
↑ 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

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

This article is issued from Wikipedia - version of the 6/15/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.