PLAYGROUND

ECUACIONES DISTRIBUCIÓN F

Definición de distribución

X∼F(df1,df2)X\sim\mathrm{F}\left(\text{df}_{1},\text{df}_{2}\right)

Dominio de distribución

x∈[0,∞)x\in [0,\infty)

Dominio y restricciones de parámetros

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

Función de distribución acumulada

FX(x)=Idf1x/(df1x+df2)(df12,df22)F_{X}\left(x\right)=I_{\text{df}_{1} x/(\text{df}_{1} x+\text{df}_{2})}\left (\tfrac{\text{df}_{1}}{2},\tfrac{\text{df}_{2}}{2}\right)

Función de densidad de probabilidad

fX(x)=(df1x)df1df2df2(df1x+df2)df1+df2x ×Beta(df12,df22)f_{X}\left(x\right)=\frac{\sqrt{\frac{(\text{df}_{1} x)^{\text{df}_{1}} \text{df}_{2}^{\text{df}_{2}}}{(\text{df}_{1} x+\text{df}_{2})^{\text{df}_{1}+\text{df}_{2}}}}}{x\ \times \text{Beta}\left(\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}

Función de punto percentil

FX−1(u)=df2×I−1(u,df12,df22)df1×(1−I−1(u,df12,df22))F^{-1}_{X}\left(u\right)=\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

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

Media paramétrica

Mean(X)=μ1′=df2df2−2if df2>2\mathrm{Mean}(X)=\mu'_{1}=\frac{\text{df}_{2}}{\text{df}_{2}-2} \quad \text{if }\text{df}_{2} > 2

Varianza paramétrica

Variance(X)=μ2′−μ1′2=2 df22 (df1+df2−2)df1(df2−2)2(df2−4)if df2>4\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\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)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=(2df1+df2−2)8(df2−4)(df2−6)df1(df1+df2−2)if df2>6\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\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)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3(8+(df2−6)×Skewness(X)2)2df2−16+3if df2>8\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{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)=df2×I−1(12,df12,df22)df1×(1−I−1(12,df12,df22))\mathrm{Median}(X)=\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)=df2(df1−2)df1(df2+2)if df1>2\mathrm{Mode}(X)=\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

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}