PLAYGROUND

ECUACIONES DISTRIBUCIÓN KUMARASWAMY EXPONENCIADA

Definición de distribución

X∼ExponentiatedKumaraswamy(α,β,λ,min,max)X\sim\mathrm{ExponentiatedKumaraswamy}\left(\alpha,\beta,\lambda,\text{min},\text{max}\right)

Dominio de distribución

x∈(min,max)x\in\left(\text{min},\text{max}\right)

Dominio y restricciones de parámetros

α∈R+,β∈R+,λ∈R+,min∈R,max∈R\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+},\lambda\in\mathbb{R}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R}

Función de distribución acumulada

FX(x)=[1−(1−z(x)α)β]λF_{X}\left(x\right)=\left[1-\left(1-z(x)^{\alpha}\right)^{\beta}\right]^{\lambda}

Función de densidad de probabilidad

fX(x)=αβλmax−min z(x)α−1(1−z(x)α)β−1[1−(1−z(x)α)β]λ−1f_{X}\left(x\right)=\frac{\alpha\beta\lambda}{\text{max}-\text{min}}\,z(x)^{\alpha-1}\left(1-z(x)^{\alpha}\right)^{\beta-1}\left[1-\left(1-z(x)^{\alpha}\right)^{\beta}\right]^{\lambda-1}

Función de punto percentil

FX−1(u)=min+(max−min)[1−(1−u1/λ)1/β]1/αF^{-1}_{X}\left(u\right)=\text{min}+\left(\text{max}-\text{min}\right)\left[1-\left(1-u^{1/\lambda}\right)^{1/\beta}\right]^{1/\alpha}

Momentos paramétricos no centrados

μ~k′=E[X~k]=λ∑j=0∞(−1)j(k/αj)Beta ⁣(jβ+1,λ)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\lambda\sum_{j=0}^{\infty}(-1)^{j}\binom{k/\alpha}{j}\text{Beta}\!\left(\tfrac{j}{\beta}+1,\lambda\right)

Media paramétrica

Mean(X)=min+(max−min)×μ~1′\mathrm{Mean}(X)=\text{min}+\left(\text{max}-\text{min}\right)\times \tilde{\mu}'_{1}

Varianza paramétrica

Variance(X)=(max−min)2(μ~2′−μ~1′2)\mathrm{Variance}(X)=\left(\text{max}-\text{min}\right)^{2}(\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)=FX−1(0.5)\mathrm{Median}(X)=F^{-1}_{X}\left(0.5\right)

Moda paramétrica

Mode(X)=arg⁡max⁡x∈(min,max)fX(x)\mathrm{Mode}(X)=\arg\max_{x\in(\text{min},\text{max})} f_{X}(x)

Información y definiciones adicionales

X~∼ExponentiatedKumaraswamy(α,β,λ,0,1)\tilde{X}\sim\mathrm{ExponentiatedKumaraswamy}\left(\alpha,\beta,\lambda,0,1\right)
z(x)=(x−min)/(max−min)z\left(x\right)=\left(x-\text{min}\right)/\left(\text{max}-\text{min}\right)
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}
(νj):Generalized binomial coefficient\binom{\nu}{j}:\text{Generalized binomial coefficient}