PLAYGROUND

ECUACIONES DISTRIBUCIÓN BETA PRIME

Definición de distribución

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

Dominio de distribución

x∈[0,∞)x\in [0,\infty)

Dominio y restricciones de parámetros

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

Función de distribución acumulada

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

Función de densidad de probabilidad

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)}

Función de punto percentil

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)}

Momentos paramétricos no centrados

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

Media paramétrica

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

Varianza paramétrica

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

Coeficiente de asimetría paramétrico

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

Curtosis paramétrica

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

Mediana paramétrica

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)}

Moda paramétrica

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

Información y definiciones adicionales

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}