Closed Form Two-Sided Bounds for Probabilities that At Least r and Exactly r Out of n Events Occur
Closed Form Two-Sided Bounds for Probabilities that At Least r and Exactly r Out of n Events Occur
复制标题
n 个事件中至少有 r 和恰好 r 发生的概率的封闭形式两侧界
DOI：
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发表时间：
1989
影响因子：
1.7
通讯作者：
A. Prékopa
中科院分区：
文献类型：
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作者：
E. Boros;A. Prékopa
在前两篇论文中，Prekopa（Prekopa，A. 1986年a。Boole-Bonferroni不等式与线性规划Oper Res. 36 145-162; Prekopa，A. 1986年b。使用线性规划对概率的精确界。出现在歌剧中。Res.）给出了n个事件中至少r个事件发生且恰好r个事件发生的概率的近似算法（1 ≤ r ≤ n）。原始和对偶线性规划问题的制定和解决的对偶型算法。本文的目的是在任意基的情况下给出基逆的封闭形式和相应的对偶向量，进而在对偶可行基的情况下给出近似概率的上、下界的封闭形式.在n个事件中至少有一个事件发生的概率近似的情况下，它表明，任何对偶向量的分量的绝对值形成一个单调递减序列。本文改进了容斥法，证明了新的概率不等式，并给出了相应的证明。
In two previous papers Prekopa (Prekopa, A. 1986a. Boole-Bonferroni inequalities and linear programming. Oper. Res. 36 145–162; Prekopa, A. 1986b. Sharp bounds on probabilities using linear programming. To appear in Oper. Res.) gave algorithms to approximate probabilities that at least r and exactly r out of n events occur (1 ≤ r ≤ n). Primal and dual linear programming problems were formulated and solved by dual type algorithms. The purpose of the present paper is to give closed forms for the basis inverse and the corresponding dual vector in case of an arbitrary basis, furthermore to give closed forms for the lower and upper bounds, approximating the probability in question, in case of a dual feasible basis. In the case when the probability that at least one out of n events occurs is approximated, it is shown that the absolute values of the components of any dual vector form a monotonically decreasing sequence. The paper improves the method of inclusion-exclusion, proves new probability inequalities and...