# F-分布

參數 概率密度函數 累積分佈函數 $d_1>0,\ d_2>0$自由度 $x \in [0; +\infty)\!$ $\frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!$ $I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)\!$ $\frac{d_2}{d_2-2}\!$ for $d_2 > 2$ $\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!$ for $d_1 > 2$ $\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!$ for $d_2 > 4$ $\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}\!$ for $d_2 > 6$ 见下文

## 定义

\begin{align} 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{align}

$F(x; d_1,d_2)=I_{\frac{d_1 x}{d_1 x + d_2}}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) ,$

$\gamma_2 = 12\frac{d_1(5d_2-22)(d_1+d_2-2)+(d_2-4)(d_2-2)^2}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}$.

## 特征

$\frac{U_1/d_1}{U_2/d_2} = \frac{U_1/U_2}{d_1/d_2}$

• 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.