PLAYGROUND

ECUACIONES DISTRIBUCIÓN BETA PRIME 4P

Definición de distribución

X∼BetaPrime4P(α,β,Loc,Sc)X\sim\mathrm{BetaPrime}_{\mathrm{4P}}\left(\alpha,\beta,\text{Loc},\text{Sc}\right)

Dominio de distribución

x∈[Loc,∞)x\in [\text{Loc},\infty)

Dominio y restricciones de parámetros

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

Función de distribución acumulada

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

Función de densidad de probabilidad

fX(x)=z(x)α−1(1+z(x))−α−βSc×Beta(α,β)f_{X}\left(x\right)=\frac{z(x)^{\alpha-1} (1+z(x))^{-\alpha -\beta}}{\text{Sc}\times \text{Beta}(\alpha,\beta)}

Función de punto percentil

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

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=Γ(k+α)Γ(β−k)Γ(α)Γ(β)if β>k\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{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

Media paramétrica

Mean(X)=Loc+Scμ~1′=Loc+Scαβ−1if β>1\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\tilde{\mu}'_{1}=\text{Loc}+\text{Sc}\frac{\alpha}{\beta-1} \quad \text{if }\beta>1

Varianza paramétrica

Variance(X)=Sc2(μ~2′−μ~1′2)=Sc2α(α+β−1)(β−2)(β−1)2if β>2\mathrm{Variance}(X)=\text{Sc}^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\text{Sc}^{2}\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2} \quad \text{if }\beta>2

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2(2α+β−1)β−3β−2α(α+β−1)if β>3\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\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

Curtosis paramétrica

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

Mediana paramétrica

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

Moda paramétrica

Mode(X)=Loc+Scα−1β+1\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\frac{\alpha-1}{\beta+1}

Información y definiciones adicionales

X~∼BetaPrime(α,β)\tilde{X}\sim \mathrm{BetaPrime}\left( \alpha,\beta \right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc}
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}