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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 Multidimensional Chebyshev's inequality ： ウィキペディア英語版 Multidimensional Chebyshev's inequality In probability theory, the multidimensional Chebyshev's inequality is a generalization of Chebyshev's inequality, which puts a bound on the probability of the event that a random variable differs from its expected value by more than a specified amount. Let ''X'' be an ''N''-dimensional random vector with expected value $\backslash mu=\backslash mathbb\; \backslash left(X\; \backslash right)$ and covariance matrix : $V=\backslash mathbb\; \backslash left(\backslash left(X\; -\; \backslash mu\; \backslash right)\; \backslash left(\; X\; -\; \backslash mu\; \backslash right)^T\; \backslash right).\; \backslash ,$ If $V$ is a positive-definite matrix, for any real number $t>0$: :$$ \mathrm\left( \sqrt > t \right) \le \frac ==Proof== Since $V$ is positive-definite, so is $V^$. Define the random variable :$$ y = \left( X-\mu\right)^T \, V^ \, \left( X-\mu\right) . Since $y$ is positive, Markov's inequality holds: :$$ \begin\mathrm\left( \sqrt > t\right) &= \mathrm\left( \sqrt > t\right)\\ &=\mathrm\left( y > t^2 \right) \\ &\le \frac .\end Finally, : &=\mathbb(\mathrm ( V^ \, \left( X-\mu\right) \, \left( X-\mu\right)^T ) )\\ &= \mathrm ( V^ V ) = N \end. 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「Multidimensional Chebyshev's inequality」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.