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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 McCullagh's parametrization of the Cauchy distributions ： ウィキペディア英語版 McCullagh's parametrization of the Cauchy distributions In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function (pdf) is :$f(x)\; =$ for ''x'' real. This has median 0, and first and third quartiles respectively −1 and +1. Generally, a Cauchy distribution is any probability distribution belonging to the same location-scale family as this one. Thus, if ''X'' has a standard Cauchy distribution and μ is any real number and σ > 0, then ''Y'' = μ + σ''X'' has a Cauchy distribution whose median is μ and whose first and third quartiles are respectively μ − σ and μ + σ. McCullagh's parametrization, introduced by Peter McCullagh, professor of statistics at the University of Chicago uses the two parameters of the non-standardised distribution to form a single complex-valued parameter, specifically, the complex number θ = μ + ''i''σ, where ''i'' is the imaginary unit. It also extends the usual range of scale parameter to include σ < 0. Although the parameter is notionally expressed using a complex number, the density is still a density over the real line. In particular the density can be written using the real-valued parameters μ and σ, which can each take positive or negative values, as :$f(x)\; =\; )\; \}\backslash ,,$ where the distribution is regarded as degenerate if σ = 0. An alternative form for the density can be written using the complex parameter θ = μ + ''i''σ as :$f(x)\; =\; \backslash ,,$ where $\backslash Im\; =\; \backslash sigma$. To the question "Why introduce complex numbers when only real-valued random variables are involved?", McCullagh wrote: In other words, if the random variable ''Y'' has a Cauchy distribution with complex parameter ''θ'', then the random variable ''Y''  * defined above has a Cauchy distribution with parameter (''aθ'' + ''b'')/(''cθ'' + ''d''). McCullagh also wrote, "The distribution of the first exit point from the upper half-plane of a Brownian particle starting at ''θ'' is the Cauchy density on the real line with parameter ''θ''." In addition, McCullagh shows that the complex-valued parameterisation allows a simple relationship to be made between the Cauchy and the "circular Cauchy distribution". ==Differential equation== McCullagh's parametrization of the pdf of the Cauchy distribution is a solution to the following differential equation: :$\backslash left\backslash$ f'(x) \left(\mu ^2+\sigma ^2+x^2-2 \mu x\right)+f(x) (2 x-2 \mu )=0, \\ f(0)=\frac+1\right)} \end\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「McCullagh's parametrization of the Cauchy distributions」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.