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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 Gaussian measure ： ウィキペディア英語版 Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the Central Limit Theorem. Loosely speaking, it states that if a random variable ''X'' is obtained by summing a large number ''N'' of independent random variables of order 1, then ''X'' is of order $\backslash sqrt$ and its law is approximately Gaussian. ==Definitions== Let ''n'' ∈ N and let ''B''0(R''n'') denote the completion of the Borel ''σ''-algebra on R''n''. Let ''λ''''n'' : ''B''0(R''n'') → (+∞ ) denote the usual ''n''-dimensional Lebesgue measure. Then the standard Gaussian measure ''γ''''n'' : ''B''0(R''n'') → (1 ) is defined by :$\backslash gamma^\; (A)\; =\; \backslash frac\}\; \backslash int\_\; \backslash exp\; \backslash left(\; -\; \backslash frac\; \backslash |\; x\; \backslash |\_\}^\; \backslash right)\; \backslash ,\; \backslash mathrm\; \backslash lambda^\; (x)$ for any measurable set ''A'' ∈ ''B''0(R''n''). In terms of the Radon–Nikodym derivative, :$\backslash frac\}\}\; (x)\; =\; \backslash frac\}\; \backslash exp\; \backslash left(\; -\; \backslash frac\; \backslash |\; x\; \backslash |\_\}^\; \backslash right).$ More generally, the Gaussian measure with mean ''μ'' ∈ R''n'' and variance ''σ''2 > 0 is given by :$\backslash gamma\_\; (A)\; :=\; \backslash frac\}\; \backslash int\_\; \backslash exp\; \backslash left(\; -\; \backslash frac^\}^\; \backslash right)\; \backslash ,\; \backslash mathrm\; \backslash lambda^\; (x).$ Gaussian measures with mean ''μ'' = 0 are known as centred Gaussian measures. The Dirac measure ''δ''''μ'' is the weak limit of $\backslash gamma\_$ as ''σ'' → 0, and is considered to be a degenerate Gaussian measure; in contrast, Gaussian measures with finite, non-zero variance are called non-degenerate Gaussian measures. 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「Gaussian measure」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.