PLAYGROUND

ECUACIONES DISTRIBUCIÓN WEIBULL 3P

Definición de distribución

X∼Weibull3P(α,Loc,β)X\sim\mathrm{Weibull_{3P}}\left(\alpha,\text{Loc},\beta\right)

Dominio de distribución

x∈[Loc,∞)x\in [\text{Loc},\infty)

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)=1−e−z(x)αF_{X}\left(x\right)=1-e^{-z(x)^\alpha}

Función de densidad de probabilidad

fX(x)=αβz(x)α−1e−z(x)αf_{X}\left(x\right)=\frac{\alpha}{\beta}z(x)^{\alpha-1}e^{-z(x)^\alpha}

Función de punto percentil

FX−1(u)=Loc+β(−ln⁡(1−u))1/αF^{-1}_{X}\left(u\right)=\text{Loc}+\beta(-\ln(1-u))^{1/\alpha}

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=βαΓ(1+kα)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\beta^\alpha \Gamma\left(1+\frac{k}{\alpha}\right)

Media paramétrica

Mean(X)=Loc+μ~1′=Loc+β Γ(1+1/α)\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}=\text{Loc}+\beta \ \Gamma(1+1/\alpha)

Varianza paramétrica

Variance(X)=μ~2′−μ~1′2=β2[Γ(1+2/α)−(Γ(1+1/α))2]\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}=\beta^2\left[\Gamma\left(1+2/\alpha\right)-\left(\Gamma\left(1+1/\alpha\right)\right)^2\right]

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+β(ln⁡(2))1/α\mathrm{Median}(X)=\text{Loc}+\beta(\ln(2))^{1/\alpha}

Moda paramétrica

Mode(X)=Loc+{β(α−1α)1/αif α>10if α≤1\mathrm{Mode}(X)=\text{Loc}+\left\{\begin{array}{cl} \beta\left(\frac{\alpha-1}{\alpha}\right)^{1/\alpha} & \text{if }\alpha>1\\ 0 & \text{if } \alpha\leq 1 \end{array} \right.

Información y definiciones adicionales

X~∼Weibull(α,β)\tilde{X}\sim\mathrm{Weibull}\left(\alpha,\beta\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
β:Scale parameter\beta:\text{Scale parameter}
z(x)=(x−Loc)/βz\left(x\right)=\left(x-\text{Loc}\right)/\beta
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}