翻訳と辞書 Words near each other ・ Hoechst AG ・ Hoechst Schering AgrEvo GmbH ・ Hoechst stain ・ Hoeckens linkage ・ Hoedads Reforestation Cooperative ・ Hoedekenskerke ・ Hoedeok Station ・ Hoedeopbap ・ Hoedjiespunt ・ Hoedown ・ Hoedown Throwdown ・ Hoedspruit ・ Hoefelman CH-1 Schatzie ・ Hoeffding ・ Hoeffding's independence test ・ Hoeffding's inequality ・ Hoeffding's lemma ・ Hoeffel ・ Hoeffer House (Syracuse, New York) ・ Hoeffken Lake ・ Hoeflea ・ Hoeflea alexandrii ・ Hoeflea anabaenae ・ Hoeflea halophila ・ Hoeflea marina ・ Hoeflea phototrophica ・ Hoeflea siderophila ・ Hoeflea suaedae ・ Hoefler ・ Hoefler & Co. Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hoeffding's inequality ： ウィキペディア英語版 Hoeffding's inequality In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of random variables deviates from its expected value. Hoeffding's inequality was proved by Wassily Hoeffding in 1963. Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality, and it is more general than the Bernstein inequality, proved by Sergei Bernstein in 1923. They are also special cases of McDiarmid's inequality. == Special case of Bernoulli random variables == Hoeffding's inequality can be applied to the important special case of identically distributed Bernoulli random variables, and this is how the inequality is often used in combinatorics and computer science. We consider a coin that shows heads with probability and tails with probability . We toss the coin times. The expected number of times the coin comes up heads is . Furthermore, the probability that the coin comes up heads at most times can be exactly quantified by the following expression: :$\backslash mathbb(\; H(n)\; \backslash leq\; k)=\; \backslash sum\_^\; \backslash binom\; p^i\; (1-p)^,$ where is the number of heads in coin tosses. When for some , Hoeffding's inequality bounds this probability by a term that is exponentially small in : :$\backslash mathbb(\; H(n)\; \backslash leq\; (p-\backslash varepsilon)\; n)\backslash leq\backslash exp\backslash left(-2\backslash varepsilon^2\; n\backslash right).$ Similarly, when for some , Hoeffding's inequality bounds the probability that we see at least more tosses that show heads than we would expect: :$\backslash mathbb(H(n)\; \backslash geq\; (p+\backslash varepsilon)n)\backslash leq\backslash exp\backslash left(-2\backslash varepsilon^2\; n\backslash right).$ Hence Hoeffding's inequality implies that the number of heads that we see is concentrated around its mean, with exponentially small tail. :$\backslash mathbb((p-\backslash epsilon)n\; \backslash leq\; H(n)\; \backslash leq\; (p+\backslash varepsilon)n)\backslash geq\; 1-2\backslash exp\backslash left(-2\backslash varepsilon^2\; n\backslash right).$ 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「Hoeffding'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.