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

參考文獻

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 (編). 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 （頁面存檔備份，存於互聯網檔案館）
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.