布尔不等式

布尔不等式可以用数学归纳法证明。

对于1个事件：

${\displaystyle P(A_{1})\leq P(A_{1})}$ 

对于n个事件：

${\displaystyle P(\bigcup _{i=_{1}}^{n}A_{i})\leq \sum _{i=_{1}}^{n}P(A_{i})}$ 

${\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)}$ 

${\displaystyle P(\bigcup _{i=_{1}}^{n+1}A_{i})=P(\bigcup _{i=_{1}}^{n}A_{i})+P(A_{n+1})-P(\bigcup _{i=_{1}}^{n}A_{i}\cap A_{n+1})}$ 

${\displaystyle P(\bigcup _{i=_{1}}^{n}A_{i}\cap A_{n+1})\geq 0,}$ 

${\displaystyle P(\bigcup _{i=_{1}}^{n+1}A_{i})\leq P(\bigcup _{i=_{1}}^{n}A_{i})+P(A_{n+1})}$ 

${\displaystyle P(\bigcup _{i=_{1}}^{n+1}A_{i})\leq \sum _{i=_{1}}^{n}P(A_{i})+P(A_{n+1})=\sum _{i=_{1}}^{n+1}P(A_{i})}$ .

令${\displaystyle A_{1},A_{2},\cdots ,A_{n}}$ 是任意概率事件。${\displaystyle X}$ 是各种事件${\displaystyle A_{i}}$ 的发生次数的随机变量。显然有：

${\displaystyle E(X)=P(A_{1})+P(A_{2})+\cdots +P(A_{n})=\sum _{i=1}^{n}P(A_{i})}$ 

因为${\displaystyle X}$ 是非负随机变量，应用马尔可夫不等式，取${\displaystyle a=1}$ ，有：

${\displaystyle P(X\geqslant 1)\leqslant E(X)=\sum _{i=1}^{n}P(A_{i})}$ 

注意到${\displaystyle P(X\geqslant 1)=P(\bigcup _{i=_{1}}^{n}A_{i})}$ 

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Bonferroni不等式 。

定义：

${\displaystyle S_{1}=\sum _{i=1}^{n}P(A_{i}),}$ 

${\displaystyle S_{2}=\sum _{1\leq i<j\leq n}P(A_{i}\cap A_{j}),}$ 

${\displaystyle S_{k}=\sum _{1\leq i_{1}<\cdots <i_{k}\leq n}P(A_{i_{1}}\cap \cdots \cap A_{i_{k}})}$ 

对于奇数k：

${\displaystyle P(\bigcup _{i=1}^{n}A_{i})\leq \sum _{j=1}^{k}(-1)^{j-1}S_{j}}$ 

对于偶数k：

${\displaystyle P(\bigcup _{i=1}^{n}A_{i})\geq \sum _{j=1}^{k}(-1)^{j-1}S_{j}}$ 

• 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）