查看“︁布尔不等式”︁的源代码

{{NoteTA |G1=Math
|1=zh-tw:布爾 ;zh-hk:布爾 ;zh-hans:布尔 ;
}}
{{Probability fundamentals}}
'''布尔不等式'''（{{lang-en|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A<sub>1</sub>、A<sub>2</sub>、A<sub>3</sub>、......：
:<math>P(\bigcup_{i} A_i) \le \sum_i P(A_i)</math>

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：
:<math>P(A_1) \le P(A_1)</math>

对于n个事件：
:<math>P(\bigcup_{i=_1}^{n} A_i) \le \sum_{i=_1}^{n} P(A_i)</math>
:<math>P(A \cup B) = P(A) + P(B) - P(A \cap B)</math>
:<math>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})</math>
:<math>P(\bigcup_{i=_1}^n A_i \cap A_{n+1}) \ge 0,</math> 
:<math>P(\bigcup_{i=_1}^{n+1} A_i) \le P(\bigcup_{i=_1}^n A_i) + P(A_{n+1})</math>
:<math>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)</math>.
==使用马尔可夫不等式的证明==
令<math>A_1,A_2,\cdots,A_n</math>是任意[[概率事件]]。<math>X</math>是各种事件<math>A_i</math>的发生次数的[[随机变量]]。显然有：
:<math>E(X)=P(A_1)+P(A_2)+\cdots+P(A_n)=\sum_{i=1}^{n}P(A_i)</math>
因为<math>X</math>是非负随机变量，应用[[馬爾可夫不等式]]，取<math>a=1</math>，有：
:<math>P(X\geqslant 1)\leqslant E(X)=\sum_{i=1}^{n}P(A_i)</math>
注意到<math>P(X\geqslant 1)=P(\bigcup_{i=_1}^{n} A_i)</math>

==邦费罗尼不等式==
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'''邦费罗尼不等式''' 。

定义：
:<math>S_1 = \sum_{i=1}^n P(A_i),</math>
:<math>S_2 = \sum_{1\le i<j\le n} P(A_i \cap A_j),</math>
:<math>S_k = \sum_{1\le i_1<\cdots<i_k\le n} P(A_{i_1}\cap \cdots \cap A_{i_k} )</math>

对于奇数k：
:<math>P( \bigcup_{i=1}^n A_i ) \le \sum_{j=1}^k (-1)^{j-1} S_j</math>

对于偶数k：
:<math>P( \bigcup_{i=1}^n A_i) \ge \sum_{j=1}^k (-1)^{j-1} S_j</math>

==参见==
*[[容斥原理]]

==参考资料==
*{{Citation
  | last = Bonferroni
  | first = Carlo E.
  | author-link = :en:Carlo Emilio Bonferroni
  | title = Teoria statistica delle classi e calcolo delle probabilità
  | journal = Pubbl. d. R. Ist. Super. di Sci. Econom. e Commerciali di Firenze
  | volume = 8
  | pages = 1–62
  | year = 1936
  | language = it
  | zbl = 0016.41103}}
* {{Citation
  | last = Dohmen
  | first = Klaus
  | title = Improved Bonferroni Inequalities via Abstract Tubes. Inequalities and Identities of Inclusion–Exclusion Type
  | place = Berlin
  | publisher = [[Springer-Verlag]]
  | series = Lecture Notes in Mathematics
  | year = 2003
  | volume = 1826
  | pages = viii+113
  | isbn = 3-540-20025-8
  | mr = 2019293
  | zbl = 1026.05009}}
* {{Citation
  | last = Galambos
  | first = János
  | author-link = :en:Janos Galambos
  | last2 = Simonelli
  | first2 = Italo
  | title = Bonferroni-Type Inequalities with Applications
  | place = New York
  | publisher = [[Springer-Verlag]]
  | series = Probability and Its Applications
  | year = 1996
  | pages = x+269
  | isbn = 0-387-94776-0
  | mr = 1402242
  | zbl = 0869.60014}}
* {{Citation
  | last = Galambos
  | first = János
  | author-link = :en:Janos Galambos
  | title = Bonferroni inequalities
  | journal = Annals of Probability
  | volume = 5
  | issue = 4
  | pages = 577–581
  | year = 1977
  | url = http://projecteuclid.org/euclid.aop/1176995765
  | doi = 10.1214/aop/1176995765
  | jstor = 2243081
  | mr = 0448478
  | zbl = 0369.60018
  | accessdate = 2014-01-12
  | archive-date = 2020-07-25
  | archive-url = https://web.archive.org/web/20200725014032/https://projecteuclid.org/euclid.aop/1176995765
  | dead-url = no
  }}

[[分类:概率不等式]]
[[Category:統計不等式]]