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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 Campbell's theorem (probability) ： ウィキペディア英語版 Campbell's theorem (probability) In probability theory and statistics, Campbell's theorem or the Campbell-Hardy theorem can refer to a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the intensity measure of the point process, which allows for the calculation of expected value and variance of the random sum. One version of the theorem, 〔D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.〕 also known as Campbell's formula, entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process.〔 There also exist equations involving moment measures and factorial moment measures that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the related fields of point processes, stochastic geometry〔 and continuum percolation theory,〔R. Meester and R. Roy. Continuum percolation, volume 119 of Cambridge tracts in mathematics, 1996.〕 spatial statistics.〔 Another result by the name of Campbell's theorem is specifically for the Poisson point process and gives a method for calculating moments as well as Laplace functionals of the Poisson point process. The name of both theorems stems from the work by Norman R. Campbell on thermionic noise, also known as shot noise, in vacuum tubes,〔 which was partly inspired by the work of Ernest Rutherford and Hans Geiger on alpha particle detection, where the Poisson point process arose as a solution to a family of differential equations by Harry Bateman.〔 In Campbell's work, he presents the moments and generating functions of the random sum of a Poisson process on the real line, but remarks that the main mathematical argument was due to G. H. Hardy, which has inspired the result to be sometimes called the Campbell-Hardy theorem.〔 ==Background== For a point process $$ defined on (''d''-dimensional) Euclidean space $\backslash textbf^d$, Campbell's theorem offers a way to calculate expectations of a function $f$ (with range in the real line $\backslash textbf$ ) defined also on $\backslash textbf^d$ and summed over $$, namely: :$E(\backslash sum\_$ is considered as a random set (see Point process notation). For a point process $$, Campbell's theorem relates the above expectation with the intensity measure ''Λ''. In relation to a Borel set ''B'' the intensity measure of $$ is defined as: :$\backslash Lambda(B)=E((B))$, where the measure notation is used such that $$ is considered a random counting measure. The quantity ''Λ(B)'' can be interpreted as the average number of points of $$ located in the set ''B''. 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「Campbell's theorem (probability)」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.