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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 Cauchy distribution ： ウィキペディア英語版 Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The simplest Cauchy distribution is called the standard Cauchy distribution. It is the distribution of a random variable that is the ratio of two independent standard normal variables and has the probability density function :$f(x;\; 0,1)\; =\; \backslash frac.\; \backslash !$ Its cumulative distribution function has the shape of an arctangent function arctan(''x''): :$F(x;\; 0,1)=\backslash frac\; \backslash arctan\backslash left(x\backslash right)+\backslash frac$ The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its mean and its variance are undefined. (But see the section ''Explanation of undefined moments'' below.) The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist.〔, Chapter 16.〕 The Cauchy distribution has no moment generating function. The Cauchy distribution $f(x;\; x\_0,\backslash gamma)$ is the distribution of the X-intercept of a ray issuing from $(x\_0,\backslash gamma)$ with a uniformly distributed angle. Its importance in physics is the result of it being the solution to the differential equation describing forced resonance.〔http://webphysics.davidson.edu/Projects/AnAntonelli/node5.html Note that the intensity, which follows the Cauchy distribution, is the square of the amplitude.〕 In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane. In spectroscopy, it is the description of the shape of spectral lines which are subject to homogeneous broadening in which all atoms interact in the same way with the frequency range contained in the line shape. Many mechanisms cause homogeneous broadening, most notably collision broadening. In its standard form, it is the maximum entropy probability distribution for a random variate ''X'' for which :$\backslash operatorname\backslash !\backslash left(\backslash right)=\backslash ln(4)$ ==History== Functions with the form of the Cauchy distribution were studied by mathematicians in the 17th century, but in a different context and under the title of the Witch of Agnesi. Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French mathematician Poisson in 1824, with Cauchy only becoming associated with it during an academic controversy in 1853.〔Cauchy and the Witch of Agnesi in ''Statistics on the Table'', S M Stigler Harvard 1999 Chapter 18〕 As such, the name of the distribution is a case of Stigler's Law of Eponymy. Poisson noted that if the mean of observations following such a distribution were taken, the mean error did not converge to any finite number. As such, Laplace's use of the Central Limit Theorem with such a distribution was inappropriate, as it assumed a finite mean and variance. Despite this, Poisson did not regard the issue as important, in contrast to Bienaymé, who was to engage Cauchy in a long dispute over the matter. 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「Cauchy 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.