F-分布

在概率论和统计学里，F-分布（F-distribution）是一种连续概率分布，^[1][2][3][4]被广泛应用于似然比率检验，特别是ANOVA中。

F分布

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

定义

如果随机变量 X 有参数为 d₁ 和 d₂ 的 F-分布，我们写作 X ~ F(d₁, d₂)。那么对于实数 x ≥ 0，X 的概率密度函数 (pdf)是

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

这里${\displaystyle \mathrm {B} }$是B函数。在很多应用中，参数 d₁ 和 d₂ 是正整数，但对于这些参数为正实数时也有定义。

累积分布函数为

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

其中 I 是正则不完全贝塔函数。

右边表格中已给出期望、方差和偏度；对于${\displaystyle d_{2}>8}$，峰度为：

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

特征

一个F-分布的随机变量是两个卡方分布变量除以自由度的比率：

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

其中：

• U₁和U₂呈卡方分布，它们的自由度（degree of freedom）分别是d₁和d₂。
• U₁和U₂是相互独立的。

参见

• F检验

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. Abramowitz, Milton; Stegun, Irene Ann (编). Chapter 26. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first. Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. 1983: 946. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642.

   LCCN 65-12253

   .
3. NIST (2006). Engineering Statistics Handbook – F Distribution （页面存档备份，存于互联网档案馆）
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.