PLAYGROUND

ECUACIONES DISTRIBUCIÓN TRAPEZOIDAL

Definición de distribución

X∼Trapezoidal(a,b,c,d)X\sim\mathrm{Trapezoidal}\left(a,b,c,d\right)

Dominio de distribución

x∈[a,d]x\in\left[a,d\right]

Dominio y restricciones de parámetros

a∈R,b∈R,c∈R,d∈R,a<b<c,b<c<da\in\mathbb{R},b\in\mathbb{R},c\in\mathbb{R},d\in\mathbb{R},a < b < c,b < c < d

Función de distribución acumulada

FX(x)={1d+c−a−b1b−a(x−a)2if  a≤x<b1d+c−a−b(2x−a−b)if  b≤x<c1−1d+c−a−b1d−c(d−x)2if  c≤x≤dF_{X}\left(x\right)=\left\{\begin{array}{cl}\frac{1}{d+c-a-b}\frac{1}{b-a}(x-a)^2 & \text{if } \ a\leq x < b \\ \frac{1}{d+c-a-b}(2x-a-b) & \text{if } \ b\leq x < c \\ 1-\frac{1}{d+c-a-b}\frac{1}{d-c}(d-x)^2 & \text{if } \ c\leq x \le d \end{array} \right.

Función de densidad de probabilidad

fX(x)={2d+c−a−bx−ab−aif  a≤x<b2d+c−a−bif  b≤x<c2d+c−a−bd−xd−cif  c≤x≤df_{X}\left(x\right)=\left\{\begin{array}{cl}\frac{2}{d+c-a-b}\frac{x-a}{b-a} & \text{if } \ a\leq x < b \\ \frac{2}{d+c-a-b} & \text{if } \ b\leq x < c \\ \frac{2}{d+c-a-b}\frac{d-x}{d-c} & \text{if } \ c\leq x \leq d \end{array} \right.

Función de punto percentil

FX−1(u)={a+u×(d+c−a−b)×(b−a)if u≤A1(a+b+u×(d+c−a−b))/2if A1≤u≤A1+A2d−(1−u)×(d+c−a−b)×(d−c)if A1+A2≤u≤A1+A2+A3F^{-1}_{X}\left(u\right)=\left\{\begin{array}{cl} a+\sqrt{u\times (d+c-a-b)\times (b-a)} & \text{if } u \leq A_{1} \\ (a+b+u\times (d+c-a-b))/2 & \text{if } A_{1} \leq u \leq A_{1}+A_{2} \\ d-\sqrt{(1-u)\times (d+c-a-b)\times (d-c)} & \text{if } A_{1}+A_{2} \leq u \leq A_{1}+A_{2}+A_{3} \end{array} \right.

Momentos paramétricos no centrados

μk′=E[Xk]=∫abxkfX(x)dx=2d+c−b−a1(k+1)(k+2)(dk+2−ck+2d−c−bk+2−ak+2b−a)\mu'_{k}=E[X^k]=\int_{a}^{b}x^{k}f_{X}\left(x\right)dx=\frac{2}{d+c-b-a}\frac{1}{(k+1)(k+2)}\left(\frac{d^{k+2}-c^{k+2}}{d-c}-\frac{b^{k+2}-a^{k+2}}{b-a}\right)

Media paramétrica

Mean(X)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Varianza paramétrica

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Curtosis paramétrica

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Mediana paramétrica

Median(X)=FX−1(1/2)\mathrm{Median}(X)=F^{-1}_{X}\left(1/2\right)

Moda paramétrica

Mode(X)∈[b,c]\mathrm{Mode}(X)\in [b,c]

Información y definiciones adicionales

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
A1=(b−a)/(d+c−a−b)A_{1}=(b-a)/(d+c-a-b)
A2=2(c−b)/(d+c−a−b)A_{2}=2(c-b)/(d+c-a-b)
A3=(d−c)/(d+c−a−b)A_{3}=(d-c)/(d+c-a-b)