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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 Boole's inequality ： ウィキペディア英語版 Boole's inequality In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. Boole's inequality is named after George Boole. Formally, for a countable set of events ''A''1, ''A''2, ''A''3, ..., we have :$\backslash biggl(\backslash bigcup\_\; A\_i\backslash biggr)\; \backslash le\; \backslash sum\_i\; (A\_i).$ In measure-theoretic terms, Boole's inequality follows from the fact that a measure (and certainly any probability measure) is ''σ''-sub-additive. ==Proof== Boole's inequality may be proved for finite collections of events using the method of induction. For the $n=1$ case, it follows that :$\backslash mathbb\; P(A\_1)\; \backslash le\; \backslash mathbb\; P(A\_1).$ For the case $n$, we have :$\backslash biggl(\backslash bigcup\_^\; A\_i\backslash biggr)\; \backslash le\; \backslash sum\_^\; (A\_i).$ Since $\backslash mathbb\; P(A\; \backslash cup\; B)\; =\; \backslash mathbb\; P(A)\; +\; \backslash mathbb\; P(B)\; -\; \backslash mathbb\; P(A\; \backslash cap\; B),$ and because the union operation is associative, we have :$\backslash biggl(\backslash bigcup\_^\; A\_i\backslash biggr)\; =\; \backslash biggl(\backslash bigcup\_^n\; A\_i\backslash biggr)\; +\; \backslash mathbb\; P(A\_)\; -\; \backslash biggl(\backslash bigcup\_^n\; A\_i\; \backslash cap\; A\_\backslash biggr).$ Since :$\backslash biggl(\backslash bigcup\_^n\; A\_i\; \backslash cap\; A\_\backslash biggr)\; \backslash ge\; 0,$ by the first axiom of probability, we have :$\backslash biggl(\backslash bigcup\_^\; A\_i\backslash biggr)\; \backslash le\; \backslash biggl(\backslash bigcup\_^n\; A\_i\backslash biggr)\; +\; \backslash mathbb\; P(A\_)$, and therefore :$\backslash biggl(\backslash bigcup\_^\; A\_i\backslash biggr)\; \backslash le\; \backslash sum\_^\; (A\_i)\; +\; \backslash mathbb\; P(A\_)\; =\; \backslash sum\_^\; (A\_i)$. 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「Boole'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.