布尔不等式 (英语: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) \le \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) \le 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) \le \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}) \ge 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) \le 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) \le \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}
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,应用马尔可夫不等式 ,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)}
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\le i<j\le 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\le i_1<\cdots<i_k\le 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 ) \le \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) \ge \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 3-540-20025-8 , 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