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Beta Prime distribution

Why it exists

The Beta prime, or beta type II, distribution describes positive ratios without the normalizing denominator used by Beta. It is natural when a variable is a quotient of two independent Gamma quantities.

Alternative names

Other names include beta type II, Pearson type VI, and, in particular contexts, the beta second-kind distribution. It is not the same as F, although F is a scaled ratio of chi-squares.

Constructions and limits

If Y and Z are independent Gamma variables with a common scale, Y/Z is Beta prime. Converting the ratio into a proportion gives a Beta variable, a useful connection for simulation and tail interpretation.

Applications

It can suit a positive imbalance measure or a ratio of two random magnitudes. For income data, validate the tail independently from the central mass: a good central fit does not guarantee a reliable extreme tail.

Before comparing means or fitting risk models, check whether the required moment exists: the tail may make the mean or variance undefined.

Beta leaves the interval and becomes odds

Transform a Beta proportion into odds and the result is Beta Prime. The inverse construction returns a Beta variable. It also appears as the ratio of two independent Gamma variables with a common scale and, after rescaling, contains the F distribution.

These origins explain the names Beta type II and inverted Beta. Its positive support has no upper bound, and the tail can be heavy. In Bayesian inference it can represent odds and certain scales, while economics uses it for ratios and sizes. Mean and variance require conditions on the second parameter; reporting either without checking them creates nonexistent summaries even when numerical fitting has converged.

Decision guide

A good candidate when: a positive variable represents odds, a ratio of Gamma quantities, or a magnitude with a power-law tail.

Compare it with: F, Lognormal, and Burr XII. Check high quantiles and moment existence: a good central fit can conceal a theoretically undefined mean or variance.

References

Beta Prime Distribution: equations and calculator

Distribution defintion

X∼BetaPrime(α,β)X\sim\mathrm{BetaPrime}\left(\alpha,\beta\right)

Distribution domain

x∈[0,∞)x\in [0,\infty)

Parameters domain and parameters constraints

α∈R+,β∈R+\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=I(x1+x,α,β)F_{X}\left(x\right)=I\left(\frac{x}{1+x},\alpha,\beta\right)

Probability density function

fX(x)=xα−1(1+x)−α−βBeta(α,β)f_{X}\left(x\right)=\frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{\text{Beta}(\alpha,\beta)}

Percent point function/Sample

FX−1(u)=I−1(u,α,β)1−I−1(u,α,β)F^{-1}_{X}\left(u\right)=\frac{I^{-1}\left(u,\alpha,\beta\right)}{1-I^{-1}\left(u,\alpha,\beta\right)}

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx=Γ(k+α)Γ(β−k)Γ(α)Γ(β)if β>k\mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{k}f_{X}\left(x\right)dx=\frac{\Gamma\left(k+\alpha\right)\Gamma\left(\beta-k\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)} \quad \text{if }\beta>k

Parametric mean

Mean(X)=μ1′=αβ−1if β>1\mathrm{Mean}(X)=\mu'_{1}=\frac{\alpha}{\beta-1} \quad \text{if }\beta>1

Parametric variance

Variance(X)=μ2′−μ1′2=α(α+β−1)(β−2)(β−1)2if β>2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2} \quad \text{if }\beta>2

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2(2α+β−1)β−3β−2α(α+β−1)if β>3\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{2(2\alpha+\beta-1)}{\beta-3}\sqrt{\frac{\beta-2}{\alpha(\alpha+\beta-1)}} \quad \text{if }\beta>3

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2if β>4\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}} \quad \text{if }\beta>4

Parametric median

Median(X)=I−1(12,α,β)1−I−1(12,α,β)\mathrm{Median}(X)=\frac{I^{-1}\left(\frac{1}{2},\alpha,\beta\right)}{1-I^{-1}\left(\frac{1}{2},\alpha,\beta\right)}

Parametric mode

Mode(X)=α−1β+1\mathrm{Mode}(X)=\frac{\alpha-1}{\beta+1}

Additional information and definitions

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
I−1(x,a,b):Inverse of regularized incomplete beta functionI^{-1}\left(x,a,b\right):\text{Inverse of regularized incomplete beta function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}