布尔不等式

布尔不等式（英語：Boole's inequality），由乔治·布尔提出，指对于全部事件的概率不大于单个事件的概率总和。 对于事件A1、A2、A3、......： P ( ⋃ i A i ) ≤ ∑ i P ( A i ) {\displaystyle P(\bigcup _{i}A_{i})\leq \sum _{i}P(A_{i})} 在测度论上，布尔不等式满足σ次可加性。 布尔不等式可以用数学归纳法证明。 对于1个事件： P ( A 1 ) ≤ P ( A 1 ) {\displaystyle P(A_{1})\leq P(A_{1})} 对于n个事件： P ( ⋃ i = 1 n A i ) ≤ ∑ i = 1 n P ( A i ) {\displaystyle P(\bigcup _{i=_{1}}^{n}A_{i})\leq \sum _{i=_{1}}^{n}P(A_{i})} P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) {\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)} P ( ⋃ i = 1 n + 1 A i ) = P ( ⋃ i = 1 n A i ) + P ( A n + 1 ) − P ( ⋃ i = 1 n A i ∩ A n + 1 ) {\displaystyle P(\bigcup _{i=_{1}}^{n+1}A_{i})=P(\bigcup _{i=_{1}}^{n}A_{i})+P(A_{n+1})-P(\bigcup _{i=_{1}}^{n}A_{i}\cap A_{n+1})} P ( ⋃ i = 1 n A i ∩ A n + 1 ) ≥ 0 , {\displaystyle P(\bigcup _{i=_{1}}^{n}A_{i}\cap A_{n+1})\geq 0,} P ( ⋃ i = 1 n + 1 A i ) ≤ P ( ⋃ i = 1 n A i ) + P ( A n + 1 ) {\displaystyle P(\bigcup _{i=_{1}}^{n+1}A_{i})\leq P(\bigcup _{i=_{1}}^{n}A_{i})+P(A_{n+1})} P ( ⋃ i = 1 n + 1 A i ) ≤ ∑ i = 1 n P ( A i ) + P ( A n + 1 ) = ∑ i = 1 n + 1 P ( A i ) {\displaystyle P(\bigcup _{i=_{1}}^{n+1}A_{i})\leq \sum _{i=_{1}}^{n}P(A_{i})+P(A_{n+1})=\sum _{i=_{1}}^{n+1}P(A_{i})} . 令 A 1 , A 2 , ⋯ , A n {\displaystyle A_{1},A_{2},\cdots ,A_{n}} 是任意概率事件。 X {\displaystyle X} 是各种事件 A i {\displaystyle A_{i}} 的发生次数的随机变量。显然有： E ( X ) = P ( A 1 ) + P ( A 2 ) + ⋯ + P ( A n ) = ∑ i = 1 n P ( A i ) {\displaystyle E(X)=P(A_{1})+P(A_{2})+\cdots +P(A_{n})=\sum _{i=1}^{n}P(A_{i})} 因为 X {\displaystyle X} 是非负随机变量，应用馬爾可夫不等式，取 a = 1 {\displaystyle a=1} ，有： P ( X ⩾ 1 ) ⩽ E ( X ) = ∑ i = 1 n P ( A i ) {\displaystyle P(X\geqslant 1)\leqslant E(X)=\sum _{i=1}^{n}P(A_{i})} 注意到 P ( X ⩾ 1 ) = P ( ⋃ i = 1 n A i ) {\displaystyle P(X\geqslant 1)=P(\bigcup _{i=_{1}}^{n}A_{i})} 布尔不等式可以推导出事件并集的上界和下界，其关系称为邦费罗尼不等式 。 定义： S 1 = ∑ i = 1 n P ( A i ) , {\displaystyle S_{1}=\sum _{i=1}^{n}P(A_{i}),} S 2 = ∑ 1 ≤ i < j ≤ n P ( A i ∩ A j ) , {\displaystyle S_{2}=\sum _{1\leq i<j\leq n}P(A_{i}\cap A_{j}),} S k = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n P ( A i 1 ∩ ⋯ ∩ A i k ) {\displaystyle S_{k}=\sum _{1\leq i_{1}<\cdots <i_{k}\leq n}P(A_{i_{1}}\cap \cdots \cap A_{i_{k}})} 对于奇数k： P ( ⋃ i = 1 n A i ) ≤ ∑ j = 1 k ( − 1 ) j − 1 S j {\displaystyle P(\bigcup _{i=1}^{n}A_{i})\leq \sum _{j=1}^{k}(-1)^{j-1}S_{j}} 对于偶数k： P ( ⋃ i = 1 n A i ) ≥ ∑ j = 1 k ( − 1 ) j − 1 S j {\displaystyle P(\bigcup _{i=1}^{n}A_{i})\geq \sum _{j=1}^{k}(-1)^{j-1}S_{j}} 容斥原理 Bonferroni, Carlo E., Teoria statistica delle classi e calcolo delle probabilità, Pubbl. d. R. Ist. Super. di Sci. Econom. e Commerciali di Firenze, 1936, 8: 1–62, Zbl 0016.41103 （意大利语）  Dohmen, Klaus, Improved Bonferroni Inequalities via Abstract Tubes. Inequalities and Identities of Inclusion–Exclusion Type, Lecture Notes in Mathematics 1826, Berlin: Springer-Verlag: viii+113, 2003, ISBN 3-540-20025-8, MR 2019293, Zbl 1026.05009  Galambos, János; Simonelli, Italo, Bonferroni-Type Inequalities with Applications, Probability and Its Applications, New York: Springer-Verlag: x+269, 1996, ISBN 0-387-94776-0, MR 1402242, Zbl 0869.60014  Galambos, János, Bonferroni inequalities, Annals of Probability, 1977, 5 (4): 577–581 [2014-01-12], JSTOR 2243081, MR 0448478, Zbl 0369.60018, doi:10.1214/aop/1176995765, （原始内容存档于2020-07-25）