PLAYGROUND

ECUACIONES DISTRIBUCIÓN GAMMA 3P

Definición de distribución

X∼Gamma3P(α,Loc,β)X\sim\mathrm{Gamma_{3P}}\left(\alpha,\text{Loc},\beta\right)

Dominio de distribución

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

Dominio y restricciones de parámetros

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

Función de distribución acumulada

FX(x)=P(α,x−Locβ)=1Γ(α)γ(α,x−Locβ)F_{X}\left(x\right)=\text{P}\left(\alpha,\frac{x-\text{Loc}}{\beta}\right)=\frac{1}{\Gamma(\alpha)} \gamma\left(\alpha,\frac{x-\text{Loc}}{\beta}\right)

Función de densidad de probabilidad

fX(x)=1Γ(α)βα(x−Loc)α−1e−x−Locβf_{X}\left(x\right)=\frac{1}{\Gamma(\alpha) \beta^\alpha} (x-\text{Loc})^{\alpha-1} e^{-\frac{x-\text{Loc}}{\beta}}

Función de punto percentil

FX−1(u)=Loc+βP−1(α,u)F^{-1}_{X}\left(u\right)=\text{Loc}+\beta \text{P}^{-1}\left(\alpha,u\right)

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=βkΓ(k+α)Γ(α)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\beta^k\frac{\Gamma(k+\alpha)}{\Gamma(\alpha)}

Media paramétrica

Mean(X)=Loc+μ~1′=Loc+αβ\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}=\text{Loc}+\alpha \beta

Varianza paramétrica

Variance(X)=μ~2′−μ~1′2=αβ2\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}=\alpha \beta^2

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2α\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}{\sqrt{\alpha}}

Curtosis paramétrica

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3+6α\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{6}{\alpha}

Mediana paramétrica

Median(X)=Loc+(α−1)βif α>1\mathrm{Median}(X)=\text{Loc}+(\alpha-1)\beta \quad \text{if }\alpha>1

Moda paramétrica

Mode(X)=Loc+βP−1(α,12)\mathrm{Mode}(X)=\text{Loc}+\beta \text{P}^{-1}\left(\alpha,\frac{1}{2}\right)

Información y definiciones adicionales

X~∼Gamma(α,β)\tilde{X}\sim\mathrm{Gamma}\left(\alpha,\beta\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
β:Scale parameter\beta:\text{Scale parameter}
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}