PLAYGROUND

ECUACIONES DISTRIBUCIÓN GUMBEL LEFT

Definición de distribución

X∼GumbelLeft(μ,σ)X\sim\mathrm{GumbelLeft}\left(\mu,\sigma\right)

Dominio de distribución

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Dominio y restricciones de parámetros

μ∈R,σ∈R+\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Función de distribución acumulada

FX(x)=1−exp⁡(−ez(x))F_{X}\left(x\right)=1-\exp\left(-e^{z(x)}\right)

Función de densidad de probabilidad

fX(x)=1σexp⁡(z(x)−ez(x))f_{X}\left(x\right)=\frac{1}{\sigma}\exp\left(z(x)-e^{z(x)}\right)

Función de punto percentil

FX−1(u)=μ+σln⁡(−ln⁡(1−u))F^{-1}_{X}\left(u\right)=\mu+\sigma\ln\left(-\ln\left(1-u\right)\right)

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫−∞∞xkfX~(x)dx\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{-\infty}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx

Media paramétrica

Mean(X)=μ+σμ~1′=μ−γσ\mathrm{Mean}(X)=\mu+\sigma\tilde{\mu}'_{1}=\mu-\gamma\sigma

Varianza paramétrica

Variance(X)=σ2(μ~2′−μ~1′2)=σ2π26\mathrm{Variance}(X)=\sigma^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\sigma^{2}\frac{\pi^{2}}{6}

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=−126ζ(3)π3\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{12\sqrt{6}\zeta(3)}{\pi^{3}}

Curtosis paramétrica

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3+125\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}}=3+\frac{12}{5}

Mediana paramétrica

Median(X)=μ+σln⁡(−ln⁡(12))\mathrm{Median}(X)=\mu+\sigma\ln\left(-\ln\left(\frac{1}{2}\right)\right)

Moda paramétrica

Mode(X)=μ\mathrm{Mode}(X)=\mu

Información y definiciones adicionales

X~∼GumbelLeft(0,1)\tilde{X}\sim\mathrm{GumbelLeft}\left(0,1\right)
μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
z(x)=(x−μ)/σz\left(x\right)=\left(x-\mu\right)/\sigma
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
γ:Euler-Mascheroni constant=0.5772156649\gamma:\text{Euler-Mascheroni constant}=0.5772156649
ζ(3):Apeˊry’s constant=1.2020569031\zeta(3):\text{Apéry's constant}=1.2020569031