PLAYGROUND

ECUACIONES DISTRIBUCIÓN NORMAL GENERALIZADA

Definición de distribución

X∼GeneralizedNormal(β,μ,α)X\sim\mathrm{GeneralizedNormal}\left(\beta,\mu,\alpha\right)

Dominio de distribución

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

Dominio y restricciones de parámetros

β∈R+,μ∈R,α∈R+\beta\in\mathbb{R}^{+},\mu\in\mathbb{R},\alpha\in\mathbb{R}^{+}

Función de distribución acumulada

FX(x)=12+sign(x−μ)2Γ(1/β)γ(1/β,∣x−μα∣β)=12+sign(x−μ)2P(1/β,∣x−μα∣β)F_{X}\left(x\right)=\frac{1}{2}+\frac{\text{sign}(x-\mu)}{2\Gamma( 1/\beta ) } \gamma\left(1/\beta,\left|\frac{x-\mu}{\alpha}\right|^\beta\right)=\frac{1}{2}+\frac{\text{sign}(x-\mu)}{2} \text{P}\left(1/\beta,\left|\frac{x-\mu}{\alpha}\right|^\beta\right)

Función de densidad de probabilidad

fX(x)=β2αΓ(1/β)exp⁡(−(∣x−μ∣α)β)f_{X}\left(x\right)=\frac{\beta}{2\alpha\Gamma(1/\beta)}\exp\left(-\left(\frac{|x-\mu|}{\alpha}\right)^\beta\right)

Función de punto percentil

FX−1(u)=sign(u−12)[αβP−1(1β,2∣u−12∣)]1/β+μF^{-1}_{X}\left(u\right)=\text{sign}(u-\frac{1}{2}) \left[\alpha^\beta \text{P}^{-1}\left(\frac{1}{\beta},2|u-\frac{1}{2}|\right)\right]^{1/\beta}+\mu

Momentos paramétricos no centrados

μk′=E[Xk]=∫−∞∞xkfX(x)dx={0if k is oddαkΓ(k+1β)/Γ(1β)if k is even\mu'_{k}=E[X^k]=\int_{-\infty}^{\infty}x^{k}f_{X}\left(x\right)dx=\left\{\begin{array}{cl}0 & \text{if }k\text{ is odd} \\ \alpha^{k} \Gamma\left(\frac{k+1}{\beta}\right) \Big/ \Gamma\left(\frac{1}{\beta}\right) & \text{if }k\text{ is even}\end{array} \right.

Media paramétrica

Mean(X)=μ+αμ1′=μ\mathrm{Mean}(X)=\mu+\alpha\mu'_{1}=\mu

Varianza paramétrica

Variance(X)=α2(μ2′−μ1′2)=α2Γ(3/β)Γ(1/β)\mathrm{Variance}(X)=\alpha^2(\mu'_{2}-\mu'^{2}_{1})=\frac{\alpha^2\Gamma(3/\beta)}{\Gamma(1/\beta)}

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=0\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=0

Curtosis paramétrica

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=Γ(5/β)Γ(1/β)Γ(3/β)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\frac{\Gamma(5/\beta)\Gamma(1/\beta)}{\Gamma(3/\beta)^2}

Mediana paramétrica

Median(X)=μ\mathrm{Median}(X)=\mu

Moda paramétrica

Mode(X)=μ\mathrm{Mode}(X)=\mu

Información y definiciones adicionales

μ:Location parameter\mu:\text{Location parameter}
α:Scale parameter\alpha:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma\left(a,x\right):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}