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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 Levy function ： ウィキペディア英語版 Lévy distribution ~~\frac}}\left(\sqrt}\right)| mean =$\backslash infty$| median =$c/2(\backslash textrm^(1/2))^2\backslash ,$, for $\backslash mu=0$| mode =$\backslash frac$, for $\backslash mu=0$| variance =$\backslash infty$| skewness =undefined| kurtosis =undefined| entropy =$\backslash frac$ where $\backslash gamma$ is Euler's constant| mgf =undefined| char =$e^\}$ In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile.〔"van der Waals profile" appears with lowercase "van" in almost all sources, such as: ''Statistical mechanics of the liquid surface'' by Clive Anthony Croxton, 1980, A Wiley-Interscience publication, ISBN 0-471-27663-4, ISBN 978-0-471-27663-0, (); and in ''Journal of technical physics'', Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995, ()〕 It is a special case of the inverse-gamma distribution. It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the normal distribution and the Cauchy distribution. ==Definition== The probability density function of the Lévy distribution over the domain $x\backslash ge\; \backslash mu$ is :$f(x;\backslash mu,c)=\backslash sqrt\}~~\backslash frac\}\}\; \backslash left(\backslash sqrt\}\backslash right)$ where $\backslash textrm(z)$ is the complementary error function. The shift parameter $\backslash mu$ has the effect of shifting the curve to the right by an amount $\backslash mu$, and changing the support to the interval [$\backslash mu$, $\backslash infty$). Like all stable distributions, the Levy distribution has a standard form f(x;0,1) which has the following property: :$f(x;\backslash mu,c)dx\; =\; f(y;0,1)dy\backslash ,$ where ''y'' is defined as :$y\; =\; \backslash frac\backslash ,$ The characteristic function of the Lévy distribution is given by :$\backslash varphi(t;\backslash mu,c)=e^~(1-i~\backslash textrm(t))\}.$ Assuming $\backslash mu=0$, the ''n''th moment of the unshifted Lévy distribution is formally defined by: :$m\_n\backslash \; \backslash stackrel\backslash \; \backslash sqrt\}\backslash int\_0^\backslash infty\; \backslash frac\}\backslash \; \backslash sqrt\}\backslash int\_0^\backslash infty\; \backslash frac\}\backslash ,dx$ which diverges for $t>0$ and is therefore not defined in an interval around zero, so that the moment generating function is not defined ''per se''. Like all stable distributions except the normal distribution, the wing of the probability density function exhibits heavy tail behavior falling off according to a power law: :$\backslash lim\_f(x;\backslash mu,c)\; =\backslash sqrt\}~\backslash frac\}\; \backslash left(-\backslash frac\backslash right)^\}\backslash right\backslash \}$ 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「Lévy 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.