# F-分布

参数 概率密度函数 累积分布函数 ${\displaystyle d_{1}>0,\ d_{2}>0}$自由度 ${\displaystyle x\in [0;+\infty )\!}$ ${\displaystyle {\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!}$ ${\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}(d_{1}/2,d_{2}/2)\!}$ ${\displaystyle {\frac {d_{2}}{d_{2}-2}}\!}$ for ${\displaystyle d_{2}>2}$ ${\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}\!}$ for ${\displaystyle d_{1}>2}$ ${\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!}$ for ${\displaystyle d_{2}>4}$ ${\displaystyle {\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!}$for ${\displaystyle d_{2}>6}$ 见下文

## 定义

{\displaystyle {\begin{aligned}f(x;d_{1},d_{2})&={\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\\&={\frac {1}{\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\left({\frac {d_{1}}{d_{2}}}\right)^{\frac {d_{1}}{2}}x^{{\frac {d_{1}}{2}}-1}\left(1+{\frac {d_{1}}{d_{2}}}\,x\right)^{-{\frac {d_{1}+d_{2}}{2}}}\end{aligned}}}

${\displaystyle F(x;d_{1},d_{2})=I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right),}$

${\displaystyle \gamma _{2}=12{\frac {d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}}$.

## 特征

${\displaystyle {\frac {U_{1}/d_{1}}{U_{2}/d_{2}}}={\frac {U_{1}/U_{2}}{d_{1}/d_{2}}}}$

• U1U2卡方分布，它们的自由度（degree of freedom）分别是d1d2
• U1U2是相互独立的。

## 参考文献

1. ^ Johnson, Norman Lloyd; Samuel Kotz; N. Balakrishnan. Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27). Wiley. 1995. ISBN 0-471-58494-0.
2. ^ Template:Abramowitz Stegun ref
3. ^ NIST (2006). Engineering Statistics Handbook – F Distribution
4. ^ Mood, Alexander; Franklin A. Graybill; Duane C. Boes. Introduction to the Theory of Statistics (Third Edition, pp. 246–249). McGraw-Hill. 1974. ISBN 0-07-042864-6.