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F-分布.
F分布
概率密度函数 ![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/74/F-distribution_pdf.svg/325px-F-distribution_pdf.svg.png) |
累积分布函数 ![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8e/F_dist_cdf.svg/325px-F_dist_cdf.svg.png) |
参数 |
自由度 |
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值域 |
![{\displaystyle x\in [0;+\infty )\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b848db9a155f453a3ee4d2179cddec82d54a13dc) |
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概率密度函数 |
![{\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)))\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65803c3bdaed5d4c035f6366343875341620b203) |
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累积分布函数 |
![{\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2))}(d_{1}/2,d_{2}/2)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9648f4a7a83c643cf3981d807bdfe317f23ec3c) |
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期望 |
for ![{\displaystyle d_{2}>2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8becc7f9a26666a2faee158c209dba7b42c4b66) |
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众数 |
for ![{\displaystyle d_{1}>2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e308c920cd72c86c6a1a8f84b0eadc3a807a711f) |
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方差 |
for ![{\displaystyle d_{2}>4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1102915ed0508d2dc5afe5bd440e7dfad0249887) |
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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)))))\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac47c2f77fbcda51696e9f0819ff405c7f4c5b47) for ![{\displaystyle d_{2}>6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cde6f4f94cb60b4cc472d49a8f614fe723590be) |
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峰度 |
见下文 |
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在概率论和统计学里,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。
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
![{\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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a55e68498f7fd2c02998e8692ce340c3e9a5cedf)
这里
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
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6e8656a1099fb6f7c20f32322e7d80415a77e6)
其中 I 是正则不完全贝塔函数。
右边表格中已给出期望、方差和偏度;对于
,峰度为:
.
特征
一个F-分布的随机变量是两个卡方分布变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))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/39135411c1de431a8edfa7091d9cd57c95393bbf)
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。
参考文献
- ^ 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.
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| 离散单变量 | 有限支集 | |
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| 无限支集 |
- beta negative binomial
- Borel
- Conway–Maxwell–Poisson
- discrete phase-type
- Delaporte
- extended negative binomial
- Flory–Schulz
- Gauss–Kuzmin
- 几何分布
- 对数分布
- mixed Poisson
- 负二项分布
- Panjer
- parabolic fractal
- 泊松分布
- Skellam
- Yule–Simon
- zeta
|
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| |
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| 连续单变量 | |
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| 混合单变量 | |
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| 联合分布 |
- Discrete:
- Ewens
- multinomial
- Continuous:
- 狄利克雷分布
- multivariate Laplace
- 多元正态分布
- multivariate stable
- multivariate t
- normal-gamma
- 随机矩阵
- LKJ
- 矩阵正态分布
- matrix t
- matrix gamma
- 威沙特分布
|
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| 定向统计 |
- 循环单变量定向统计
- 圆均匀分布
- univariate von Mises
- wrapped normal
- wrapped Cauchy
- wrapped exponential
- wrapped asymmetric Laplace
- wrapped Lévy
- 球形双变量
- Kent
- 环形双变量
- bivariate von Mises
- 多变量
- von Mises–Fisher
- Bingham
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| 退化分布和奇异分布 | |
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| 其它 |
- Circular
- 复合泊松分布
- elliptical
- exponential
- natural exponential
- location–scale
- Maximum entropy
- Mixture
- Pearson
- Tweedie
- Wrapped
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