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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク F distribution ： ウィキペディア英語版 F-distribution } ,\frac\right)}\!| cdf =$I\_\}\; \backslash left(\backslash tfrac,\; \backslash tfrac\; \backslash right)$| mean =$\backslash frac\backslash !$ for ''d''2 > 2| median =| mode =$\backslash frac\backslash ;\backslash frac$ for ''d''1 > 2| variance =$\backslash frac\backslash !$ for ''d''2 > 4| skewness =$\backslash frac\}\backslash !$ for ''d''2 > 6| kurtosis =''see text''| entropy =| mgf =''does not exist, raw moments defined in text and in 〔〔 ''| char =''see text''|}} The ''F''-distribution, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is, in probability theory and statistics, a continuous probability distribution.〔NIST (2006). (Engineering Statistics Handbook – F Distribution )〕 The ''F''-distribution arises frequently as the null distribution of a test statistic, most notably in the analysis of variance; see ''F''-test. ==Definition== If a random variable ''X'' has an ''F''-distribution with parameters ''d''1 and ''d''2, we write ''X'' ~ F(''d''1, ''d''2). Then the probability density function (pdf) for ''X'' is given by :$$ \begin f(x; d_1,d_2) &= \frac} } ,\frac\right)} \\ &=\frac,\frac\right)} \left(\frac\right)^} x^ - 1} \left(1+\frac\,x\right)^} \end for real ''x'' ≥ 0. Here $\backslash mathrm$ is the beta function. In many applications, the parameters ''d''1 and ''d''2 are positive integers, but the distribution is well-defined for positive real values of these parameters. The cumulative distribution function is :$F(x;\; d\_1,d\_2)=I\_\}\backslash left\; (\backslash tfrac,\; \backslash tfrac\; \backslash right)\; ,$ where ''I'' is the regularized incomplete beta function. The expectation, variance, and other details about the F(''d''1, ''d''2) are given in the sidebox; for ''d''2 > 8, the excess kurtosis is :$\backslash gamma\_2\; =\; 12\backslash frac$. The ''k''-th moment of an F(''d''1, ''d''2) distribution exists and is finite only when 2''k'' < ''d''2 and it is equal to 〔(【引用サイトリンク】 first1 = Marco )〕 :$\backslash mu\; \_(k)\; =\backslash left(\; \backslash frac\}\backslash right)^\backslash frac+k\backslash right)\; \}\backslash right)\; \}\backslash frac-k\backslash right)\; \}\backslash right)\; \}$ 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 is listed incorrectly in many standard references (e.g., 〔). The correct expression 〔Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," ''Biometrika'', 69: 261–264 〕 is :$\backslash varphi^F\_(s)\; =\; \backslash frac)\})\}\; U\; \backslash !\; \backslash left(\backslash frac,1-\backslash frac,-\backslash frac\; \backslash imath\; s\; \backslash right)$ where ''U''(''a'', ''b'', ''z'') is the confluent hypergeometric function of the second kind. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「F-distribution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.