PLAYGROUND

ECUACIONES DISTRIBUCIÓN PERT

Definición de distribución

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

Dominio de distribución

x∈[a,c]x\in\left[a,c\right]

Dominio y restricciones de parámetros

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

Función de distribución acumulada

FX(x)=I(z(x),α1,α2)F_{X}\left(x\right)=I(z(x),\alpha_{1},\alpha_{2})

Función de densidad de probabilidad

fX(x)=(x−a)α1−1(c−x)α2−1Beta(α1,α2)(c−a)α1+α2−1f_{X}\left(x\right)=\frac{(x-a)^{\alpha_{1}-1}(c-x)^{\alpha_{2}-1}} {\text{Beta}(\alpha_{1},\alpha_{2})(c-a)^{\alpha_{1}+\alpha_{2}-1}}

Función de punto percentil

FX−1(u)=a+(c−a)⋅I−1(u,α1,α2)F^{-1}_{X}\left(u\right)=a+(c-a)\cdot I^{-1}\left(u,\alpha_{1},\alpha_{2}\right)

Momentos paramétricos no centrados

μk′=E[Xk]=∫acxkfX(x)dx\mu'_{k}=E[X^k]=\int_{a}^{c}x^{k}f_{X}\left(x\right)dx

Media paramétrica

Mean(X)=μ1′=a+4b+c6\mathrm{Mean}(X)=\mu'_{1}=\frac{a+4b+c}{6}

Varianza paramétrica

Variance(X)=μ2′−μ1′2=(Mean(X)−a)(c−Mean(X))7\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{(\mathrm{Mean}(X)-a)(c-\mathrm{Mean}(X))}{7}

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2 (α2−α1)α1+α2+1(α1+α2+2)α1α2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{2\,(\alpha_{2}-\alpha_{1})\sqrt{\alpha_{1}+\alpha_{2}+1}}{(\alpha_{1}+\alpha_{2}+2)\sqrt{\alpha_{1}\alpha_{2}}}

Curtosis paramétrica

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=6[(α1−α2)2(α1+α2+1)−α1α2(α1+α2+2)]α1α2(α1+α2+2)(α1+α2+3)+3\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}}=\frac{6[(\alpha_{1}-\alpha_{2})^2 (\alpha_{1} +\alpha_{2}+1)-\alpha_{1} \alpha_{2} (\alpha_{1}+\alpha_{2}+2)]}{\alpha_{1} \alpha_{2} (\alpha_{1}+\alpha_{2}+2) (\alpha_{1}+\alpha_{2}+3)}+3

Mediana paramétrica

Median(X)=a+(c−a)⋅I−1(12,α1,α2)\mathrm{Median}(X)=a+(c-a)\cdot I^{-1}\left(\frac{1}{2},\alpha_{1},\alpha_{2}\right)

Moda paramétrica

Mode(X)=b\mathrm{Mode}(X)=b

Información y definiciones adicionales

z(x)=(x−a)/(c−a)z\left(x\right)=\left(x-a\right)/\left(c-a\right)
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
α1=4b+c−5ac−a,α2=5c−a−4bc−a\alpha_{1}=\frac{4b+c-5a} {c-a},\alpha_{2}=\frac{5c-a-4b} {c-a}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
I−1(x,a,b):Inverse of regularized incomplete beta functionI^{-1}\left(x,a,b\right):\text{Inverse of regularized incomplete beta function}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}