F-分佈

在機率論和統計學裏，F-分佈（F-distribution）是一種連續機率分佈，^[1][2][3][4]被廣泛應用於似然比率檢驗，特別是ANOVA中。

F分佈

機率密度函數
累積分佈函數
參數	${\displaystyle d_{1}>0,\ d_{2}>0}$自由度
值域	${\displaystyle x\in [0;+\infty )\!}$
機率密度函數	${\displaystyle {\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!}$
累積分佈函數	${\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}(d_{1}/2,d_{2}/2)\!}$
期望值	${\displaystyle {\frac {d_{2}}{d_{2}-2}}\!}$ for ${\displaystyle d_{2}>2}$
眾數	${\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}\!}$ for ${\displaystyle d_{1}>2}$
變異數	${\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!}$ 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	見下文

定義

如果隨機變量 X 有參數為 d₁ 和 d₂ 的 F-分佈，我們寫作 X ~ F(d₁, d₂)。那麼對於實數 x ≥ 0，X 的機率密度函數 (pdf)是

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

這裏${\displaystyle \mathrm {B} }$是B函數。在很多應用中，參數 d₁ 和 d₂ 是正整數，但對於這些參數為正實數時也有定義。

累積分佈函數為

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

其中 I 是正則不完全貝塔函數。

右邊表格中已給出期望值、方差和偏度；對於${\displaystyle d_{2}>8}$，峰度為：

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

特徵

一個F-分佈的隨機變量是兩個卡方分佈變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呈卡方分佈，它們的自由度（degree of freedom）分別是d₁和d₂。
• U₁和U₂是相互獨立的。

參見

• F檢驗