PLAYGROUND

ECUACIONES DISTRIBUCIÓN KUMARASWAMY

Definición de distribución

X∼Kumaraswamy(α,β,min,max)X\sim\mathrm{Kumaraswamy}\left(\alpha,\beta,\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+,min∈R,max∈R\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R}

Función de distribución acumulada

FX(x)=1−(1−z(x)α)βF_{X}\left(x\right)=1-(1-z(x)^\alpha)^\beta

Función de densidad de probabilidad

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

Función de punto percentil

FX−1(u)=min+(max−min)×(1−(1−u)1β)1αF^{-1}_{X}\left(u\right)=\text{min}+\left(\text{max}-\text{min}\right)\times (1-(1-u)^\frac{1}{\beta})^\frac{1}{\alpha}

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫01xkfX~(x)dx=βBeta(1+kα,β)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{1}x^{k}f_{\tilde{X}}\left(x\right)dx=\beta \text{Beta}(1+\frac{k}{\alpha},\beta)

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)=min+(max−min)×(1−2−1/b)1/a\mathrm{Median}(X)=\text{min}+\left(\text{max}-\text{min}\right)\times\left(1-2^{-1/b}\right)^{1/a}

Moda paramétrica

Mode(X)=min+(max−min)×(a−1ab−1)1/a\mathrm{Mode}(X)=\text{min}+\left(\text{max}-\text{min}\right)\times\left(\frac{a-1}{ab-1}\right)^{1/a}

Información y definiciones adicionales

X~∼Kumaraswamy(α,β,0,1)\tilde{X}\sim\mathrm{Kumaraswamy}\left(\alpha,\beta,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}