在概率论 和统计学 裡,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
概率密度函數
累積分布函數
参数
d
1
>
0
,
d
2
>
0
{\displaystyle d_{1}>0,\ d_{2}>0}
自由度 值域
x
∈
[
0
;
+
∞
)
{\displaystyle x\in [0;+\infty )\!}
概率密度函数
(
d
1
x
)
d
1
d
2
d
2
(
d
1
x
+
d
2
)
d
1
+
d
2
x
B
(
d
1
2
,
d
2
2
)
{\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)}}\!}
累積分布函數
I
d
1
x
d
1
x
+
d
2
(
d
1
/
2
,
d
2
/
2
)
{\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}(d_{1}/2,d_{2}/2)\!}
期望值
d
2
d
2
−
2
{\displaystyle {\frac {d_{2}}{d_{2}-2}}\!}
for
d
2
>
2
{\displaystyle d_{2}>2}
眾數
d
1
−
2
d
1
d
2
d
2
+
2
{\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}\!}
for
d
1
>
2
{\displaystyle d_{1}>2}
方差
2
d
2
2
(
d
1
+
d
2
−
2
)
d
1
(
d
2
−
2
)
2
(
d
2
−
4
)
{\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!}
for
d
2
>
4
{\displaystyle d_{2}>4}
偏度
(
2
d
1
+
d
2
−
2
)
8
(
d
2
−
4
)
(
d
2
−
6
)
d
1
(
d
1
+
d
2
−
2
)
{\displaystyle {\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
{\displaystyle d_{2}>6}
峰度
见下文
如果随机变量 X 有参数为 d 1 和 d 2 的 F -分布,我们写作 X ~ F(d 1 , d 2 )。那么对于实数 x ≥ 0,X 的概率密度函数 (pdf)是
f
(
x
;
d
1
,
d
2
)
=
(
d
1
x
)
d
1
d
2
d
2
(
d
1
x
+
d
2
)
d
1
+
d
2
x
B
(
d
1
2
,
d
2
2
)
=
1
B
(
d
1
2
,
d
2
2
)
(
d
1
d
2
)
d
1
2
x
d
1
2
−
1
(
1
+
d
1
d
2
x
)
−
d
1
+
d
2
2
{\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
B
{\displaystyle \mathrm {B} }
是B函数 。在很多应用中,参数 d 1 和 d 2 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
d
1
x
d
1
x
+
d
2
(
d
1
2
,
d
2
2
)
,
{\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值 、方差 和偏度 ;对于
d
2
>
8
{\displaystyle d_{2}>8}
,峰度 为:
γ
2
=
12
d
1
(
5
d
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
)
{\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)}}}
.
一个F -分布的随机变量 是两个卡方分佈 变量除以自由度的比率:
U
1
/
d
1
U
2
/
d
2
=
U
1
/
U
2
d
1
/
d
2
{\displaystyle {\frac {U_{1}/d_{1}}{U_{2}/d_{2}}}={\frac {U_{1}/U_{2}}{d_{1}/d_{2}}}}
其中:
U 1 和U 2 呈卡方分佈 ,它们的自由度 (degree of freedom)分别是d 1 和d 2 。
U 1 和U 2 是相互独立的。
^ Johnson, Norman Lloyd; Samuel Kotz; N. Balakrishnan. Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27). Wiley. 1995. ISBN 0-471-58494-0 .
^ 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 . .
^ NIST (2006). Engineering Statistics Handbook – F Distribution (页面存档备份 ,存于互联网档案馆 )
^ 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 .