# F-分布

參數 $d_1>0,\ d_2>0$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

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

• 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。

$\frac{12(20d_2-8d_2^2+d_2^3+44d_1-32d_1d_2+5d_2^2d_1-22d_1^2+5d_2d_1^2-16)}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}$

$g(x) = \frac{1}{\mathrm{B}(d_1/2, d_2/2)} \; \left(\frac{d_1\,x}{d_1\,x + d_2}\right)^{d_1/2} \; \left(1-\frac{d_1\,x}{d_1\,x + d_2}\right)^{d_2/2} \; x^{-1}$

$G(x) = I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)$

I是不完全Beta函数。

## F分配定義式

$F\left(U_1 , U_2 \right) = \frac {\frac {\chi^2 \left( U_1 \right)} {U_1} } {\frac {\chi^2 \left( U_2 \right)} {U_2} } \sim F\left(U_1 , U_2 \right)$