PLAYGROUND

ECUACIONES DISTRIBUCIÓN TOPP-LEONE

Definición de distribución

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

Función de distribución acumulada

FX(x)=[z(x)(2−z(x))]αF_{X}\left(x\right)=\left[z(x)\left(2-z(x)\right)\right]^{\alpha}

Función de densidad de probabilidad

fX(x)=2αmax−min(1−z(x))[z(x)(2−z(x))]α−1f_{X}\left(x\right)=\frac{2\alpha}{\text{max}-\text{min}}\left(1-z(x)\right)\left[z(x)\left(2-z(x)\right)\right]^{\alpha-1}

Función de punto percentil

FX−1(u)=min+(max−min)(1−1−u1/α)F^{-1}_{X}\left(u\right)=\text{min}+\left(\text{max}-\text{min}\right)\left(1-\sqrt{1-u^{1/\alpha}}\right)

Momentos paramétricos no centrados

μ~k′=E[X~k]=∑i=0k(ki)(−1)iα Beta ⁣(α,1+i2)\tilde{\mu}'_{k}=E[\tilde{X}^{k}]=\sum_{i=0}^{k}\binom{k}{i}\left(-1\right)^{i}\alpha\,\text{Beta}\!\left(\alpha,1+\tfrac{i}{2}\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)=min+(max−min)(1−1−0.51/α)\mathrm{Median}(X)=\text{min}+\left(\text{max}-\text{min}\right)\left(1-\sqrt{1-0.5^{1/\alpha}}\right)

Moda paramétrica

Mode(X)=min+(max−min)(1−α−12α−1)\mathrm{Mode}(X)=\text{min}+\left(\text{max}-\text{min}\right)\left(1-\sqrt{\tfrac{\alpha-1}{2\alpha-1}}\right)

Información y definiciones adicionales

X~∼ToppLeone(α,0,1)\tilde{X}\sim\mathrm{ToppLeone}\left(\alpha,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}
Mode formula valid when α>1, otherwise Mode(X)=min\text{Mode formula valid when } \alpha>1,\text{ otherwise }\mathrm{Mode}(X)=\text{min}