PLAYGROUND

ECUACIONES DISTRIBUCIÓN F 4P

Definición de distribución

X∼F4P(df1,df2,Loc,Sc)X\sim\mathrm{F_{4P}}\left(\text{df}_{1},\text{df}_{2},\text{Loc},\text{Sc}\right)

Dominio de distribución

x∈[Loc,∞)x\in\left[\text{Loc},\infty\right)

Dominio y restricciones de parámetros

df1∈R+,df2∈R+,Loc∈R,Sc∈R+\text{df}_{1}\in\mathbb{R}^{+},\text{df}_{2}\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}

Función de distribución acumulada

FX(x)=Idf1z(x)/(df1z(x)+df2)(df12,df22)F_{X}\left(x\right)=I_{\text{df}_{1} z(x)/(\text{df}_{1} z(x)+\text{df}_{2})}\left (\tfrac{\text{df}_{1}}{2},\tfrac{\text{df}_{2}}{2}\right)

Función de densidad de probabilidad

fX(x)=1Sc×(df1z(x))df1df2df2(df1z(x)+df2)df1+df2z(x) Beta(df12,df22)f_{X}\left(x\right)=\frac{1}{\text{Sc}}\times \frac{\sqrt{\frac{(\text{df}_{1} z(x))^{\text{df}_{1}} \text{df}_{2}^{\text{df}_{2}}}{(\text{df}_{1} z(x)+\text{df}_{2})^{\text{df}_{1}+\text{df}_{2}}}}}{z(x)\,\text{Beta}\left(\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}

Función de punto percentil

FX−1(u)=Loc+Scdf2×I−1(u,df12,df22)df1×(1−I−1(u,df12,df22))F^{-1}_{X}\left(u\right)=\text{Loc}+\text{Sc}\frac{\text{df}_{2}\times I^{-1}\left(u,\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}{\text{df}_{1}\times \left(1-I^{-1}\left(u,\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)\right)}

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=Γ(df12+k)Γ(df12)Γ(df22−k)Γ(df22)(df2df1)kif df2>2k\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{\Gamma\left(\tfrac{\text{df}_{1}}{2}+k\right) }{\Gamma\left(\tfrac{\text{df}_{1}}{2}\right)}\frac{\Gamma\left(\tfrac{\text{df}_{2}}{2}-k\right) }{\Gamma\left(\tfrac{\text{df}_{2}}{2}\right) }\left(\frac{\text{df}_{2}}{\text{df}_{1}}\right)^k \quad \text{if }\text{df}_{2} > 2k

Media paramétrica

Mean(X)=Loc+Scμ~1′=Loc+Scdf2df2−2if df2>2\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\tilde{\mu}'_{1}=\text{Loc}+\text{Sc}\frac{\text{df}_{2}}{\text{df}_{2}-2} \quad \text{if }\text{df}_{2} > 2

Varianza paramétrica

Variance(X)=Sc2(μ~2′−μ~1′2)=Sc22 df22 (df1+df2−2)df1(df2−2)2(df2−4)if df2>4\mathrm{Variance}(X)=\text{Sc}^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\text{Sc}^{2}\frac{2\,\text{df}_{2}^2\,(\text{df}_{1}+\text{df}_{2}-2)}{\text{df}_{1} (\text{df}_{2}-2)^2 (\text{df}_{2}-4)} \quad \text{if }\text{df}_{2} > 4

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=(2df1+df2−2)8(df2−4)(df2−6)df1(df1+df2−2)if df2>6\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 \text{df}_{1}+\text{df}_{2}-2) \sqrt{8 (\text{df}_{2}-4)}}{(\text{df}_{2}-6) \sqrt{\text{df}_{1} (\text{df}_{1}+\text{df}_{2} -2)}}\quad \text{if }\text{df}_{2} > 6

Curtosis paramétrica

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3(8+(df2−6)×Skewness(X)2)2df2−16+3if df2>8\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}}=\frac{3\left(8+\left(\text{df}_{2}-6\right)\times \mathrm{Skewness}(X)^{2}\right)}{2\text{df}_{2}-16}+3\quad \text{if }\text{df}_{2} > 8

Mediana paramétrica

Median(X)=Loc+Scdf2×I−1(12,df12,df22)df1×(1−I−1(12,df12,df22))\mathrm{Median}(X)=\text{Loc}+\text{Sc}\frac{\text{df}_{2}\times I^{-1}\left(\frac{1}{2},\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}{\text{df}_{1}\times \left(1-I^{-1}\left(\frac{1}{2},\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)\right)}

Moda paramétrica

Mode(X)=Loc+Scdf2(df1−2)df1(df2+2)if df1>2\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\frac{\text{df}_{2}\left(\text{df}_{1}-2\right)}{\text{df}_{1}\left(\text{df}_{2}+2\right)} \quad \text{if }\text{df}_{1} > 2

Información y definiciones adicionales

X~∼F(df1,df2)\tilde{X}\sim\mathrm{F}\left(\text{df}_{1},\text{df}_{2}\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc}
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}