PLAYGROUND

ECUACIONES DISTRIBUCIÓN LOGLOGISTICA 3P

Definición de distribución

X∼LogLogistic3P(Loc,α,β)X\sim\mathrm{LogLogistic_{3P}}\left(\text{Loc},\alpha,\beta\right)

Dominio de distribución

x∈[Loc,∞)x\in [\text{Loc},\infty)

Dominio y restricciones de parámetros

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

Función de distribución acumulada

FX(x)=11+((x−Loc)/α)−βF_{X}\left(x\right)=\frac{1}{1+((x-\text{Loc})/\alpha)^{-\beta}}

Función de densidad de probabilidad

fX(x)=(β/α)((x−Loc)/α)β−1(1+((x−Loc)/α)β)2f_{X}\left(x\right)=\frac{ (\beta/\alpha)((x-\text{Loc})/\alpha)^{\beta-1} }{ \left (1+((x-\text{Loc})/\alpha)^{\beta}\right)^2 }

Función de punto percentil

FX−1(u)=Loc+α(u1−u)1/βF^{-1}_{X}\left(u\right)=\text{Loc}+\alpha\left(\frac{u}{1-u}\right)^{1/\beta}

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=αkBeta(1−k/β,1+k/β)=αk kπ/βsin⁡(kπ/β)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\alpha^k \text{Beta}(1-k/\beta,1+k/\beta)=\alpha^k\,\frac{k\pi/\beta}{\sin(k\pi/\beta)}

Media paramétrica

Mean(X)=Loc+μ~1′\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}

Varianza paramétrica

Variance(X)=μ~2′−μ~1′2\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5\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}}

Curtosis paramétrica

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2\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}}

Mediana paramétrica

Median(X)=Loc+α\mathrm{Median}(X)=\text{Loc}+\alpha

Moda paramétrica

Mode(X)=Loc+α(β−1β+1)1/β\mathrm{Mode}(X)=\text{Loc}+\alpha\left(\frac{\beta-1}{\beta+1}\right)^{1/\beta}

Información y definiciones adicionales

X~∼LogLogistic(α,β)\tilde{X}\sim\mathrm{LogLogistic}\left(\alpha,\beta\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
α:Scale parameter\alpha:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}