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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 Fréchet distribution ： ウィキペディア英語版 Fréchet distribution | cdf =$e^)^\}$ | mean =$\backslash begin$ \ m+s\Gamma\left(1-\frac\right) & \text \alpha>1 \\ \ \infty & \text \end | median =$m+\backslash frac\backslash right)^$| variance = $\backslash begin$ \ s^2\left(\Gamma\left(1-\frac\right)- \left(\Gamma\left(1-\frac\right)\right)^2\right) & \text \alpha>2 \\ \ \infty & \text \end | skewness = $\backslash begin$ \ \frac\right)-3\Gamma\left(1-\frac \right)\Gamma\left(1-\frac \right)+2\Gamma^3\left(1-\frac \right)}\right)-\Gamma^2\left(1-\frac\right) \right)^3 }} & \text \alpha>3 \\ \ \infty & \text \end | g_k =| kurtosis = $\backslash begin$ \ -6+ \frac\right) -4\Gamma\left(1-\frac\right) \Gamma\left(1-\frac\right)+3 \Gamma^2\left(1-\frac \right)} \right) - \Gamma^2 \left(1-\frac\right) \right )^2} & \text \alpha>4 \\ \ \infty & \text \end | | entropy =$1\; +\; \backslash frac\; +\; \backslash gamma\; +\backslash ln\; \backslash left(\; \backslash frac\; \backslash right)$, where $\backslash gamma$ is the Euler–Mascheroni constant.| mgf = 〔 Note: Moment $k$ exists if $\backslash alpha>k$ | char = 〔 | }} The Fréchet distribution, also known as inverse Weibull distribution,〔〔 is a special case of the generalized extreme value distribution. It has the cumulative distribution function :$\backslash Pr(X\; \backslash le\; x)=e^\; x>0.$ where ''α'' > 0 is a shape parameter. It can be generalised to include a location parameter ''m'' (the minimum) and a scale parameter ''s'' > 0 with the cumulative distribution function :$\backslash Pr(X\; \backslash le\; x)=e^\backslash right)^\}\; \backslash text\; x>m.$ Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958. ==Characteristics== The single parameter Fréchet with parameter $\backslash alpha$ has standardized moment :$\backslash mu\_k=\backslash int\_0^\backslash infty\; x^k\; f(x)dx=\backslash int\_0^\backslash infty\; t^\}e^\; \backslash ,\; dt,$ (with $t=x^$) defined only for : :$\backslash mu\_k=\backslash Gamma\backslash left(1-\backslash frac\backslash right)$ where $\backslash Gamma\backslash left(z\backslash right)$ is the Gamma function. In particular: * For $\backslash alpha>1$ the expectation is $E()=\backslash Gamma(1-\backslash tfrac)$ * For $\backslash alpha>2$ the variance is $\backslash text(X)=\backslash Gamma(1-\backslash tfrac)-\backslash big(\backslash Gamma(1-\backslash tfrac)\backslash big)^2.$ The quantile $q\_y$ of order $y$ can be expressed through the inverse of the distribution, :$q\_y=F^(y)=\backslash left(-\backslash log\_e\; y\; \backslash right)^\}$. In particular the median is: :$q\_=(\backslash log\_e\; 2)^\}.$ The mode of the distribution is $\backslash left(\backslash frac\backslash right)^\backslash frac.$ Especially for the 3-parameter Fréchet, the first quartile is $q\_1=\; m+\backslash frac)\}\}.$ Also the quantiles for the mean and mode are: :$F(mean)=\backslash exp\; \backslash left(\; -\backslash Gamma^\; \backslash left(1-\; \backslash frac\; \backslash right)\; \backslash right)$ :$F(mode)=\backslash exp\; \backslash left(\; -\backslash frac\; \backslash right).$ 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「Fréchet 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.