In this paper sharp bounds for the probability of union of n arbitrary events following unknown probability distribution are found, when only \(m(m < n)\) out of n binomial moments of the events are given. The bounds are found using a class of special matrices and their inverses. Generalized closed form probabilistic bounds are found for the probability of occurrence of at least r out of n events by using the same class of matrices and their inverses. In numerical examples section the parameter r is studied and tables are given to depict the effect of r over the sharp bounds. As an application, MAXSAT problem is also discussed and the effect of using relatively higher binomial moments over the chance of having the optimal upper bound less than 1 is discussed.

中文翻译：

---

当已知m个二项式矩时，n个事件的并集概率的鲜明边界

在本文中，当仅给出事件的n个二项式矩中的\（m（m <n）\）时，发现了n个任意事件遵循未知概率分布的并集概率的清晰边界。使用一类特殊矩阵及其逆来找到边界。通过使用相同类别的矩阵及其逆，找到n个事件中至少r个发生概率的广义封闭形式概率边界。在数值示例部分中，研究了参数r并给出了表以描述r的影响越过界限。作为应用，还讨论了MAXSAT问题，并讨论了使用相对较高的二项式矩对最佳上限小于1的机会的影响。