PLAYGROUND

ECUACIONES DISTRIBUCIÓN ALPHA

Definición de distribución

X∼Alpha(α,Loc,Sc)X\sim\mathrm{Alpha}\left(\alpha,\text{Loc},\text{Sc}\right)

Dominio de distribución

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

Dominio y restricciones de parámetros

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

Función de distribución acumulada

FX(x)=Φ(α−1z(x))Φ(α)F_{X}\left(x\right)=\frac{\Phi\left(\alpha-\frac{1}{z(x)}\right)}{\Phi\left(\alpha\right)}

Función de densidad de probabilidad

fX(x)=1Sc⋅z(x)2⋅Φ(α)⋅2πexp⁡(−12(α−1z(x))2)f_{X}\left(x\right)=\frac{1}{\text{Sc}\cdot z(x)^{2}\cdot \Phi\left(\alpha\right)\cdot\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\alpha-\frac{1}{z(x)}\right)^{2}\right)

Función de punto percentil

FX−1(u)=Loc+Sc×1α−Φ−1(uΦ(α))F_{X}^{-1}\left(u\right)=\text{Loc}+\text{Sc}\times \frac{1}{\alpha-\Phi^{-1}\left(u\Phi\left(\alpha\right)\right)}

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫0∞xkfX~(x)dx\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx

Media paramétrica

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

Varianza paramétrica

Variance(X)=Sc2⋅(μ~2′−μ~1′2)\mathrm{Variance}(X)=\text{Sc}^{2}\cdot(\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+Scα−Φ−1(12Φ(α))\mathrm{Median}(X)=\text{Loc}+\frac{\text{Sc}}{\alpha-\Phi^{-1}\left(\frac{1}{2}\Phi\left(\alpha\right)\right)}

Moda paramétrica

Mode(X)=Loc+Sc(α2+8−α)4\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\frac{(\sqrt{\alpha^{2}+8}-\alpha)}{4}

Información y definiciones adicionales

X~∼Alpha(α,0,1)\tilde{X}\sim \mathrm{Alpha}\left(\alpha,0,1\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz(x)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}