PLAYGROUND

ECUACIONES DISTRIBUCIÓN PARETO SEGUNDO TIPO

Definición de distribución

X∼ParetoSecondKind(xm,α,Loc)X\sim\mathrm{ParetoSecondKind}\left(x_\mathrm{m},\alpha,\text{Loc}\right)

Dominio de distribución

x∈(Loc,∞)x\in\left(\text{Loc},\infty\right)

Dominio y restricciones de parámetros

xm∈R+,α∈R+,Loc∈Rx_\mathrm{m}\in\mathbb{R}^{+},\alpha\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R}

Función de distribución acumulada

FX(x)=1−[1+x−Locxm]−αF_{X}\left(x\right)=1-\left[{1+\frac{x-\text{Loc}}{x_\mathrm{m}}}\right]^{-\alpha}

Función de densidad de probabilidad

fX(x)=αxm[1+x−Locxm]−(α+1)f_{X}\left(x\right)=\frac{\alpha}{x_\mathrm{m}} \left[{1+\frac{x-\text{Loc}}{x_\mathrm{m}}}\right]^{-(\alpha+1)}

Función de punto percentil

FX−1(u)=Loc+xm[(1−p)−1α−1]F^{-1}_{X}\left(u\right)=\text{Loc}+x_\mathrm{m} \left[\left(1-p\right)^{-\frac{1}{\alpha}} -1\right]

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=xmkΓ(α−k)Γ(1+k)Γ(α)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{x_\mathrm{m}^k \Gamma(\alpha-k)\Gamma(1+k)}{\Gamma(\alpha)}

Media paramétrica

Mean(X)=μ~1′=xmα−1if α>1\mathrm{Mean}(X)=\tilde{\mu}'_{1}=\frac{x_\mathrm{m}}{{\alpha -1}} \quad \text{if }\alpha>1

Varianza paramétrica

Variance(X)=μ~2′−μ~1′2=xm2α(α−1)2(α−2)if α>2\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}=\frac{x_\mathrm{m}^2 \alpha}{(\alpha-1)^2(\alpha-2)} \quad \text{if }\alpha>2

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2(1+α)α−3 α−2αif α>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(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}} \quad \text{if }\alpha>3

Curtosis paramétrica

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=6(α3+α2−6α−2)α(α−3)(α−4)if α>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}}=\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)} \quad \text{if }\alpha>4

Mediana paramétrica

Median(X)=xm(2α−1)\mathrm{Median}(X)=x_\mathrm{m}\left(\sqrt[\alpha]{2}-1\right)

Moda paramétrica

Mode(X)=0\mathrm{Mode}(X)=0

Información y definiciones adicionales

X∼ParetoSecondKind(xm,α,0)X\sim\mathrm{ParetoSecondKind}\left(x_\mathrm{m},\alpha,0\right)
xm:Scale parameterx_\mathrm{m}:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}