PLAYGROUND

ECUACIONES DISTRIBUCIÓN ERLANG 3P

Definición de distribución

X∼Erlang3P(k,β,Loc)X\sim\mathrm{Erlang_{3P}}\left(k,\beta,\text{Loc}\right)

Dominio de distribución

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

Dominio y restricciones de parámetros

k∈N+,β∈R+,Loc∈Rk\in\mathbb{N}^{+},\beta\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R}

Función de distribución acumulada

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

Función de densidad de probabilidad

fX(x)=(x−Loc)k−1e−x−Locββk(k−1)!f_{X}\left(x\right)=\frac{(x-\text{Loc})^{k-1} e^{-\frac{x-\text{Loc}}{\beta}}}{\beta^k(k-1)!}

Función de punto percentil

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

Momentos paramétricos no centrados

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

Media paramétrica

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

Varianza paramétrica

Variance(X)=μ~2′−μ~1′2\mathrm{Variance}(X)=\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+P(k,12β)\mathrm{Median}(X)=\text{Loc}+\text{P}(k,\frac{1}{2\beta})

Moda paramétrica

Mode(X)=Loc+β⋅(k−1)\mathrm{Mode}(X)=\text{Loc}+\beta\cdot\left(k-1\right)

Información y definiciones adicionales

X~∼Erlang(k,β)\tilde{X}\sim\mathrm{Erlang}\left(k,\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}