PLAYGROUND

ECUACIONES DISTRIBUCIÓN PARETO PRIMER TIPO

Definición de distribución

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

Dominio de distribución

x∈[Loc+xm,∞)x\in [\text{Loc}+x_\mathrm{m},\infty)

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−(xmx−Loc)αF_{X}\left(x\right)=1-\left(\frac{x_\mathrm{m}}{x-\text{Loc}}\right)^\alpha

Función de densidad de probabilidad

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

Función de punto percentil

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

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫xm∞xkfX~(x)dx={∞if α≤kαxmkα−kif α>k\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{x_\mathrm{m}}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\left\{\begin{array}{cl}\infty & \text{if } \alpha\le k \\ \frac{\alpha x_\mathrm{m}^k}{\alpha-k} & \text{if } \alpha>k \end{array} \right.

Media paramétrica

Mean(X)=Loc+μ~1′=Loc+αxmα−1if α>1\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}=\text{Loc}+\dfrac{\alpha 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})=\dfrac{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−3α−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)=Loc+xm2α\mathrm{Median}(X)=\text{Loc}+x_\mathrm{m} \sqrt[\alpha]{2}

Moda paramétrica

Mode(X)=Loc+xm\mathrm{Mode}(X)=\text{Loc}+x_\mathrm{m}

Información y definiciones adicionales

X~∼ParetoFirstKind(xm,α,0)\tilde{X}\sim\mathrm{ParetoFirstKind}\left(x_\mathrm{m},\alpha,0\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
xm:Scale parameterx_\mathrm{m}:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}