F-分布

F分布

概率密度函數
累積分佈函數
參數	$d_{1}>0,\ d_{2}>0$$d_{1}>0,\ d_{2}>0$自由度
支撑集	$x\in [0;+\infty )\!$$x\in [0;+\infty )\!$
概率密度函数	${\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)}}\!$${\frac {{\sqrt {{\frac {(d_{1}\,x)^{{d_{1}}}\,\,d_{2}^{{d_{2}}}}{(d_{1}\,x+d_{2})^{{d_{1}+d_{2}}}}}}}}{x\,{\mathrm {B}}\!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!$
累積分佈函數	$I_{\frac {d_{1}x}{d_{1}x+d_{2}}}(d_{1}/2,d_{2}/2)\!$$I_{{{\frac {d_{1}x}{d_{1}x+d_{2}}}}}(d_{1}/2,d_{2}/2)\!$
期望值	${\frac {d_{2}}{d_{2}-2}}\!$${\frac {d_{2}}{d_{2}-2}}\!$ for $d_{2}>2$$d_{2}>2$
眾數	${\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}\!$${\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}\!$ for $d_{1}>2$$d_{1}>2$
方差	${\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!$${\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!$ for $d_{2}>4$$d_{2}>4$
偏度	${\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!$${\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!$ for $d_{2}>6$$d_{2}>6$
峰度	见下文

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

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

${\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}}$${\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$\mathrm {B}$${\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

$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),$$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值、方差和偏度；对于$d_{2}>8$$d_{2}>8$，峰度为：

$\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)}}$$\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的比率：

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