查看“F-分布”的源代码

{{noteTA
|G1=Math
}}

{{機率分佈
|name = F分布
|type = 密度
|pdf_image = [[Image:F-distribution pdf.svg|325px]]|
|cdf_image = [[Image:F_dist_cdf.svg|325px]]|
|parameters =<math>d_1>0,\ d_2>0</math>自由度
|support = <math>x \in [0; +\infty)\!</math>
|pdf = <math>\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)}\!</math>
|cdf = <math>I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)\!</math>
|mean = <math>\frac{d_2}{d_2-2}\!</math> for <math>d_2 > 2</math>
|median =
|mode = <math>\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!</math> for <math>d_1 > 2</math>
|variance = <math>\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!</math> for <math>d_2 > 4</math>
|skewness = <math>\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}\!</math><br />for <math>d_2 > 6</math>
|kurtosis = ''见下文''
|entropy =
|mgf =
|char =
}}

在[[概率论]]和[[统计学]]里，'''''F''-分布'''（''F''-distribution）是一种[[概率分布|连续概率分布]]，<ref name=johnson>{{cite book  | last = Johnson
  | first = Norman Lloyd
  |author2=Samuel Kotz |author3=N. Balakrishnan
   | title = Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27)
  | publisher = Wiley
  | year = 1995
  | isbn = 0-471-58494-0}}</ref><ref name=abramowitz>{{Abramowitz_Stegun_ref|26|946}}</ref><ref>NIST (2006).  [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3665.htm Engineering Statistics Handbook – F Distribution]</ref><ref>{{cite book  | last = Mood
  | first = Alexander
  |author2=Franklin A. Graybill |author3=Duane C. Boes
   | title = Introduction to the Theory of Statistics (Third Edition, pp. 246–249)
  | publisher = McGraw-Hill
  | year = 1974
  | isbn = 0-07-042864-6}}</ref>被广泛应用于[[似然比率检验]]，特别是[[方差分析|ANOVA]]中。
== 定义 ==
如果[[随机变量]] ''X'' 有参数为 ''d''<sub>1</sub> 和 ''d''<sub>2</sub> 的 ''F''-分布，我们写作 ''X'' ~ F(''d''<sub>1</sub>, ''d''<sub>2</sub>)。那么对于实数 ''x'' ≥ 0，''X'' 的[[概率密度函数]] (pdf)是

:<math> 
\begin{align}
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{align}
</math>

这里<math>\mathrm{B}</math>是[[B函数]]。在很多应用中，参数 ''d''<sub>1</sub> 和 ''d''<sub>2</sub> 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

:<math>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) ,</math>

其中 ''I'' 是[[B函数#不完全贝塔函数|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<math>d_2>8</math>，[[峰度]]为：

:<math>\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)}</math>.
<!--
The ''k''-th moment of an F(''d''<sub>1</sub>, ''d''<sub>2</sub>) distribution exists and is finite only when 2''k'' < ''d''<sub>2</sub> and it is equal to <ref name=taboga>{{cite web | last1 = Taboga | first1 = Marco | url = http://www.statlect.com/F_distribution.htm | title = The F distribution}}</ref>

:<math>\mu _{X}(k) =\left( \frac{d_{2}}{d_{1}}\right)^{k}\frac{\Gamma \left(\tfrac{d_1}{2}+k\right) }{\Gamma \left(\tfrac{d_1}{2}\right) }\frac{\Gamma \left(\tfrac{d_2}{2}-k\right) }{\Gamma \left( \tfrac{d_2}{2}\right) }</math>

The ''F''-distribution is a particular parametrization of the [[beta prime distribution]], which is also called the beta distribution of the second kind.

The [[Characteristic function (probability theory)|characteristic function]] is listed incorrectly in many standard references (e.g., <ref name=abramowitz />). The correct expression <ref>Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," ''[[Biometrika]]'', 69: 261–264  {{jstor|2335882}}</ref> is

:<math>\varphi^F_{d_1, d_2}(s) = \frac{\Gamma(\frac{d_1+d_2}{2})}{\Gamma(\tfrac{d_2}{2})} U \! \left(\frac{d_1}{2},1-\frac{d_2}{2},-\frac{d_2}{d_1} \imath s \right)</math>

where ''U''(''a'', ''b'', ''z'') is the [[confluent hypergeometric function]] of the second kind.-->

== 特征 ==
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：

:<math>
\frac{U_1/d_1}{U_2/d_2}
=
\frac{U_1/U_2}{d_1/d_2}

</math>

其中：

* ''U''<sub>''1''</sub>和''U''<sub>2</sub>呈[[卡方分佈]]，它们的[[自由度 (统计学)|自由度]]（degree of freedom）分别是''d''<sub>''1''</sub>和''d''<sub>2</sub>。
* ''U''<sub>1</sub>和''U''<sub>2</sub>是相互独立的。

== 参考文献 ==
{{reflist}}

{{概率分布类型列表}}

[[Category:统计学]]
[[Category:连续分布]]