PLAYGROUND

ECUACIONES DISTRIBUCIÓN BRADFORD

Definición de distribución

X∼Bradford(c,min,max)X\sim\mathrm{Bradford}\left(c,\text{min},\text{max}\right)

Dominio de distribución

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

Dominio y restricciones de parámetros

c∈R+,min∈R,max∈R,min<maxc\in\mathbb{R}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R},\text{min} < \text{max}

Función de distribución acumulada

FX(x)=ln⁡(1+c⋅z(x))kF_{X}\left(x\right)=\frac{\ln\left(1+c\cdot z(x)\right)}{k}

Función de densidad de probabilidad

fX(x)=ck(1+c⋅z(x))(max−min)f_{X}\left(x\right)=\frac{c}{k\left(1+c\cdot z(x)\right)\left(\text{max}-\text{min}\right)}

Función de punto percentil

FX−1(u)=min+(max−min)×(1+c)u−1cF^{-1}_{X}\left(u\right)=\text{min}+(\text{max}-\text{min})\times \frac{\left(1+c\right)^{u}-1}{c}

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫01xkfX~(x)dx\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{1}x^{k}f_{\tilde{X}}\left(x\right)dx

Media paramétrica

Mean(X)=min+(max−min)⋅μ~1′=min+(max−min)⋅c−kck\mathrm{Mean}(X)=\text{min}+\left(\text{max}-\text{min}\right)\cdot\tilde{\mu}'_{1}=\text{min}+\left(\text{max}-\text{min}\right)\cdot\frac{c-k}{ck}

Varianza paramétrica

Variance(X)=(max−min)2⋅(μ~2′−μ~1′2)=(max−min)2⋅(c+2)k−2c2ck2\mathrm{Variance}(X)=\left(\text{max}-\text{min}\right)^{2}\cdot(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\left(\text{max}-\text{min}\right)^{2}\cdot\frac{\left(c+2\right)k-2c}{2ck^{2}}

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2(12c2−9kc(c+2)+2k2(c(c+3)+3))c(c(k−2)+2k)(3c(k−2)+6k)\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}}=\frac{\sqrt{2}\left(12c^{2}-9kc\left(c+2\right)+2k^{2}\left(c\left(c+3\right)+3\right)\right)}{\sqrt{c\left(c\left(k-2\right)+2k\right)}\left(3c\left(k-2\right)+6k\right)}

Curtosis paramétrica

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3+c3(k−3)(k(3k−16)+24)+12kc2(k−4)(k−3)+6ck2(3k−14)+12k33c(c(k−2)+2k)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}}=3+\frac{c^{3}\left(k-3\right)\left(k\left(3k-16\right)+24\right)+12kc^{2}\left(k-4\right)\left(k-3\right)+6ck^{2}\left(3k-14\right)+12k^{3}}{3c\left(c\left(k-2\right)+2k\right)^{2}}

Mediana paramétrica

Median(X)=min+(max−min)⋅(1+c)frac12−1c\mathrm{Median}(X)=\text{min}+(\text{max}-\text{min})\cdot\frac{\left(1+c\right)^{frac{1}{2}}-1}{c}

Moda paramétrica

Mode(X)=min\mathrm{Mode}(X)=\text{min}

Información y definiciones adicionales

X~∼Bradford(c,0,1)\tilde{X}\sim\mathrm{Bradford}\left(c,0,1\right)
k=ln⁡(1+c)k=\ln(1+c)
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}