PLAYGROUND

ECUACIONES DISTRIBUCIÓN BETA

Definición de distribución

X∼Beta(α,β,A,B)X\sim\mathrm{Beta}\left(\alpha,\beta,A,B\right)

Dominio de distribución

x∈(A,B)x\in\left(A,B\right)

Dominio y restricciones de parámetros

α∈R+,β∈R+,A∈R,B∈R,A<B\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+},A\in\mathbb{R},B\in\mathbb{R},A < B

Función de distribución acumulada

FX(x)=I(z(x),α,β)F_{X}\left(x\right)=I\left(z(x),\alpha,\beta\right)

Función de densidad de probabilidad

fX(x)=z(x)α−1(1−z(x))β−1Beta(α,β)(B−A)f_{X}\left(x\right)=\frac{z(x)^{\alpha-1}\left(1-z(x)\right)^{\beta-1}}{\text{Beta}(\alpha,\beta)(B-A)}

Función de punto percentil

FX−1(u)=A+(B−A)×I−1(u,α,β)F^{-1}_{X}\left(u\right)=A+(B-A)\times I^{-1}\left(u,\alpha,\beta\right)

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)=A+(B−A)⋅μ~1′=A+α(B−A)α+β\mathrm{Mean}(X)=A+\left(B-A\right)\cdot\tilde{\mu}'_{1}=A+\frac{\alpha\left(B-A\right)}{\alpha+\beta}

Varianza paramétrica

Variance(X)=(B−A)2⋅(μ~2′−μ~1′2)=αβ(B−A)2(α+β)2(α+β+1)\mathrm{Variance}(X)=\left(B-A\right)^{2}\cdot(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\frac{\alpha\beta\left(B-A\right)^{2}}{(\alpha+\beta)^2(\alpha+\beta+1)}

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2 (β−α)α+β+1(α+β+2)αβ\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{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}

Curtosis paramétrica

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3+6[(α−β)2(α+β+1)−αβ(α+β+2)]αβ(α+β+2)(α+β+3)\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{6[(\alpha-\beta)^2 (\alpha +\beta+1)-\alpha \beta (\alpha+\beta+2)]}{\alpha \beta (\alpha+\beta+2) (\alpha+\beta+3)}

Mediana paramétrica

Median(X)=A+(B−A)×I−1(12,α,β)if α,β>1\mathrm{Median}(X)=A+(B-A)\times I^{-1}\left(\frac{1}{2},\alpha,\beta\right) \quad \text{if }\alpha,\beta>1

Moda paramétrica

Mode(X)=A+(B−A)α−1α+β−2if α,β>1\mathrm{Mode}(X)=A+(B-A)\frac{\alpha-1}{\alpha+\beta-2} \quad \text{if }\alpha,\beta>1

Información y definiciones adicionales

X~∼Beta(α,β,0,1)\tilde{X}\sim\mathrm{Beta}\left(\alpha,\beta,0,1\right)
z(x)=(x−A)/(B−A)z\left(x\right)=\left(x-A\right)/\left(B-A\right)
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
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}