PLAYGROUND

ECUACIONES DISTRIBUCIÓN LOGGAMMA

Definición de distribución

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

Dominio de distribución

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

Dominio y restricciones de parámetros

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

Función de distribución acumulada

FX(x)=γ(c,ex)Γ(c)=P(c,ez(x))F_{X}\left(x\right)=\frac{\gamma\left(c,e^{x}\right)}{\Gamma\left(c\right)}=\text{P}\left(c,e^{z(x)}\right)

Función de densidad de probabilidad

fX(x)=exp⁡(cz(x)−ez(x))σΓ(c)f_{X}\left(x\right)=\frac{\exp\left(cz(x)-e^{z(x)}\right)}{\sigma\Gamma\left(c\right)}

Función de punto percentil

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

Momentos paramétricos no centrados

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

Media paramétrica

Mean(X)=μ1′=μ+σψ0\mathrm{Mean}(X)=\mu'_{1}=\mu+\sigma\psi_{0}

Varianza paramétrica

Variance(X)=μ2′−μ1′2=α2ψ1(c)\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\alpha^{2}\psi_{1}\left(c\right)

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=ψ2(c)ψ1(c)\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{\psi_{2}\left(c\right)}{\psi_{1}\left(c\right)}

Curtosis paramétrica

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=ψ3(c)ψ1(c)\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{\psi_{3}\left(c\right)}{\psi_{1}\left(c\right)}

Mediana paramétrica

Median(X)=μ+σln⁡(P−1(1/2,c))\mathrm{Median}(X)=\mu+\sigma\ln\left(\text{P}^{-1}\left(1/2,c\right)\right)

Moda paramétrica

Mode(X)=μ+σln⁡(c)\mathrm{Mode}(X)=\mu+\sigma\ln(c)

Información y definiciones adicionales

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}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}\left(a,x\right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
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}
ψ0(x):Digamma function\psi_{0}\left(x\right):\text{Digamma function}
ψn(x):Polygamma function of order n∈N\psi_{n}\left(x\right):\text{Polygamma function of order }n\in\mathbb{N}