Beta prime distribution

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Beta prime

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \alpha >0}${\displaystyle \alpha >0} shape (real) ${\displaystyle \beta >0}${\displaystyle \beta >0} shape (real)
Support	${\displaystyle x\in [0,\infty )\!}${\displaystyle x\in [0,\infty )\!}
PDF	${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}\!}${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}\!}
CDF	${\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}}${\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}} where ${\displaystyle I_{x}(\alpha ,\beta )}${\displaystyle I_{x}(\alpha ,\beta )} is the regularized incomplete beta function
Mean	${\displaystyle {\frac {\alpha }{\beta -1}}}${\displaystyle {\frac {\alpha }{\beta -1}}} if ${\displaystyle \beta >1}${\displaystyle \beta >1}
Mode	${\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}${\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}
Variance	${\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}${\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}} if ${\displaystyle \beta >2}${\displaystyle \beta >2}
Skewness	${\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}}${\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}} if ${\displaystyle \beta >3}${\displaystyle \beta >3}
Excess kurtosis	${\displaystyle 6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}}${\displaystyle 6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}} if ${\displaystyle \beta >4}${\displaystyle \beta >4}
Entropy	${\displaystyle {\begin{aligned}&\log \left(\mathrm {B} (\alpha ,\beta )\right)+(\alpha -1)(\psi (\beta )-\psi (\alpha ))\\+&(\alpha +\beta )\left(\psi (1-\alpha -\beta )-\psi (1-\beta )+{\frac {\pi \sin(\alpha \pi )}{\sin(\beta \pi )\sin((\alpha +\beta )\pi ))}}\right)\end{aligned}}}${\displaystyle {\begin{aligned}&\log \left(\mathrm {B} (\alpha ,\beta )\right)+(\alpha -1)(\psi (\beta )-\psi (\alpha ))\\+&(\alpha +\beta )\left(\psi (1-\alpha -\beta )-\psi (1-\beta )+{\frac {\pi \sin(\alpha \pi )}{\sin(\beta \pi )\sin((\alpha +\beta )\pi ))}}\right)\end{aligned}}} where ${\displaystyle \psi }${\displaystyle \psi } is the digamma function.
MGF	Does not exist
CF	${\displaystyle {\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{,2,0円}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|,円-it\right)}${\displaystyle {\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{,2,0円}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|,円-it\right)}

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. If ${\displaystyle p\in [0,1]}${\displaystyle p\in [0,1]} has a beta distribution, then the odds ${\displaystyle {\frac {p}{1-p}}}${\displaystyle {\frac {p}{1-p}}} has a beta prime distribution.

Beta prime distribution is defined for ${\displaystyle x>0}${\displaystyle x>0} with two parameters α and β, having the probability density function:

${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}}${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}}

where B is the Beta function.

The cumulative distribution function is

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

where I is the regularized incomplete beta function.

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

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

${\displaystyle E[X^{k}]={\frac {\mathrm {B} (\alpha +k,\beta -k)}{\mathrm {B} (\alpha ,\beta )}}.}${\displaystyle E[X^{k}]={\frac {\mathrm {B} (\alpha +k,\beta -k)}{\mathrm {B} (\alpha ,\beta )}}.}

For ${\displaystyle k\in \mathbb {N} }${\displaystyle k\in \mathbb {N} } with ${\displaystyle k<\beta ,}${\displaystyle k<\beta ,} this simplifies to

${\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}${\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}

The cdf can also be written as

${\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot \mathrm {B} (\alpha ,\beta )}}}${\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot \mathrm {B} (\alpha ,\beta )}}}

where ${\displaystyle {}_{2}F_{1}}${\displaystyle {}_{2}F_{1}} is the Gauss's hypergeometric function ₂F₁ .

The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters (^[2] 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.

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

• ${\displaystyle p>0}${\displaystyle p>0} shape (real)
• ${\displaystyle q>0}${\displaystyle q>0} scale (real)

having the probability density function:

${\displaystyle f(x;\alpha ,\beta ,p,q)={\frac {p\left({\frac {x}{q}}\right)^{\alpha p-1}\left(1+\left({\frac {x}{q}}\right)^{p}\right)^{-\alpha -\beta }}{q\mathrm {B} (\alpha ,\beta )}}}${\displaystyle 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 }}{q\mathrm {B} (\alpha ,\beta )}}}

with mean

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

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

This generalization can be obtained via the following invertible transformation. If ${\displaystyle y\sim \beta '(\alpha ,\beta )}${\displaystyle y\sim \beta '(\alpha ,\beta )} and ${\displaystyle x=qy^{1/p}}${\displaystyle x=qy^{1/p}} for ${\displaystyle q,p>0}${\displaystyle q,p>0}, then ${\displaystyle x\sim \beta '(\alpha ,\beta ,p,q)}${\displaystyle x\sim \beta '(\alpha ,\beta ,p,q)}.

The compound gamma distribution^[3] 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:

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

where ${\displaystyle G(x;a,b)}${\displaystyle G(x;a,b)} is the gamma pdf with shape ${\displaystyle a}${\displaystyle a} and inverse scale ${\displaystyle b}${\displaystyle b}.

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.

Another way to express the compounding is if ${\displaystyle r\sim G(\beta ,q)}${\displaystyle r\sim G(\beta ,q)} and ${\displaystyle x\mid r\sim G(\alpha ,r)}${\displaystyle x\mid r\sim G(\alpha ,r)}, then ${\displaystyle x\sim \beta '(\alpha ,\beta ,1,q)}${\displaystyle x\sim \beta '(\alpha ,\beta ,1,q)}. This gives one way to generate random variates with compound gamma, or beta prime distributions. Another is via the ratio of independent gamma variates, as shown below.

• If ${\displaystyle X\sim \beta '(\alpha ,\beta )}${\displaystyle X\sim \beta '(\alpha ,\beta )} then ${\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )}${\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )}.
• If ${\displaystyle Y\sim \beta '(\alpha ,\beta )}${\displaystyle Y\sim \beta '(\alpha ,\beta )}, and ${\displaystyle X=qY^{1/p}}${\displaystyle X=qY^{1/p}}, then ${\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}${\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}.
• If ${\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}${\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)} then ${\displaystyle kX\sim \beta '(\alpha ,\beta ,p,kq)}${\displaystyle kX\sim \beta '(\alpha ,\beta ,p,kq)}.
• ${\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )}${\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )}

• If ${\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )}${\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )}, then ${\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}${\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}. This property can be used to generate beta prime distributed variates.
• If ${\displaystyle X\sim \beta '(\alpha ,\beta )}${\displaystyle X\sim \beta '(\alpha ,\beta )}, then ${\displaystyle {\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )}${\displaystyle {\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )}. This is a corollary from the property above.
• If ${\displaystyle X\sim F(2\alpha ,2\beta )}${\displaystyle X\sim F(2\alpha ,2\beta )} has an F-distribution, then ${\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )}${\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )}, or equivalently, ${\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})}${\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})}.
• For gamma distribution parametrization I:
  • If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})}${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})} are independent, then ${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}. Note ${\displaystyle \theta _{1},\theta _{2},{\tfrac {\theta _{1}}{\theta _{2}}}}${\displaystyle \theta _{1},\theta _{2},{\tfrac {\theta _{1}}{\theta _{2}}}} are all scale parameters for their respective distributions.
• For gamma distribution parametrization II:
  • If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})}${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})} are independent, then ${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}. The ${\displaystyle \beta _{k}}${\displaystyle \beta _{k}} are rate parameters, while ${\displaystyle {\tfrac {\beta _{2}}{\beta _{1}}}}${\displaystyle {\tfrac {\beta _{2}}{\beta _{1}}}} is a scale parameter.
  • If ${\displaystyle \beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})}${\displaystyle \beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})} and ${\displaystyle X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})}${\displaystyle X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})}, then ${\displaystyle X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})}${\displaystyle X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})}. The ${\displaystyle \beta _{k}}${\displaystyle \beta _{k}} are rate parameters for the gamma distributions, but ${\displaystyle \beta _{1}}${\displaystyle \beta _{1}} is the scale parameter for the beta prime.
• ${\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)}${\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)} the Dagum distribution
• ${\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)}${\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)} the Singh–Maddala distribution.
• ${\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )}${\displaystyle \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 ${\displaystyle x_{m}}${\displaystyle x_{m}} and shape parameter ${\displaystyle \alpha }${\displaystyle \alpha }, then ${\displaystyle {\dfrac {X}{x_{m}}}-1\sim \beta ^{\prime }(1,\alpha )}${\displaystyle {\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 ${\displaystyle \alpha }${\displaystyle \alpha } and scale parameter ${\displaystyle \lambda }${\displaystyle \lambda }, then ${\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}${\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}.
• If X has a standard Pareto Type IV distribution with shape parameter ${\displaystyle \alpha }${\displaystyle \alpha } and inequality parameter ${\displaystyle \gamma }${\displaystyle \gamma }, then ${\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )}${\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )}, or equivalently, ${\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)}${\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)}.
• The inverted Dirichlet distribution is a generalization of the beta prime distribution.
• If ${\displaystyle X\sim \beta '(\alpha ,\beta )}${\displaystyle X\sim \beta '(\alpha ,\beta )}, then ${\displaystyle \ln X}${\displaystyle \ln X} has a generalized logistic distribution. More generally, if ${\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}${\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}, then ${\displaystyle \ln X}${\displaystyle \ln X} has a scaled and shifted generalized logistic distribution.
• If ${\displaystyle X\sim \beta '\left({\frac {1}{2}},{\frac {1}{2}}\right)}${\displaystyle X\sim \beta '\left({\frac {1}{2}},{\frac {1}{2}}\right)}, then ${\displaystyle \pm {\sqrt {X}}}${\displaystyle \pm {\sqrt {X}}} follows a Cauchy distribution, which is equivalent to a student-t distribution with the degrees of freedom of 1.

• 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

• MathWorld article

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