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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 Paley–Zygmund inequality ： ウィキペディア英語版 Paley–Zygmund inequality In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its mean and variance (i.e., its first two moments). The inequality was proved by Raymond Paley and Antoni Zygmund. Theorem: If ''Z'' ≥ 0 is a random variable with finite variance, and if $0\; \backslash le\; \backslash theta\; \backslash le\; 1$, then :$$ \operatorname( Z > \theta\operatorname() ) \ge (1-\theta)^2 \frac. Proof: First, :$$ \operatorname() = \operatorname + \operatorname. The first addend is at most $\backslash theta\; \backslash operatorname()$, while the second is at most $\backslash operatorname()^\; \backslash operatorname(\; Z\; >\; \backslash theta\backslash operatorname())^$ by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎ == Related inequalities == The Paley–Zygmund inequality can be written as :$$ \operatorname( Z > \theta \operatorname() ) \ge \frac()^2}. This can be improved. By the Cauchy–Schwarz inequality, :$$ \operatorname) \mathbf_} ] \le \operatorname^ \operatorname( Z > \theta \operatorname() )^ which, after rearranging, implies that :$$ \operatorname(Z > \theta \operatorname()) \ge \frac()^2}()^2}. This inequality is sharp; equality is achieved if Z almost surely equals a positive constant, for example. 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「Paley–Zygmund 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.