布尔不等式 （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 。
证明 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})}
使用马尔可夫不等式的证明 令
A
1
,
A
2
,
⋯
,
A
n
{\displaystyle A_{1},A_{2},\cdots ,A_{n}}
概率事件 。
X
{\displaystyle X}
A
i
{\displaystyle A_{i}}
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}
马尔科夫不等式 ，取
a
=
1
{\displaystyle a=1}
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})}
Bonferroni不等式 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：
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}}
参见 参考资料 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 3-540-20025-8MR 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 0-387-94776-0MR 1402242 Zbl 0869.60014 Galambos, János , Bonferroni inequalities , Annals of Probability, 1977, 5 (4): 577–581, JSTOR 2243081 MR 0448478 Zbl 0369.60018 doi:10.1214/aop/1176995765 
