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Unified Approach to Probability Problems and Estimation Algorithms Associated With Symmetric Functions

机译：与对称函数相关的概率问题和估计算法的统一方法

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In this article we introduce a simple powerful methodology where we replace the independent variables λ1, ..., λn in various symmetric functions as well as in Vieta’s formulas by the indication functions of the events Ai , i = 1, ..., n, i.e., λi = 1(Ai), i = 1, ..., n. Both the random variable K that counts the number of events that actually occurred and the proposed obvious identity Π n i=1 (z?1(Ai)) ≡ (z?1)K z n?K that solely depends on K play a central role in this article. Just by choosing different values for z (real, complex, and random) and taking expectations of the various functions we provide other simple proofs of known results as well as obtain new results. The estimation algorithms for computing the expected elementary symmetric functions via least squares based on IFFT in the complex domain (z ∈ C) and least squares or linear programming in the real domain (z ∈ R) are noteworthy. Similarly, we we use Newton’s identities and some well known inequalities to obtain new results and inequalities. Then, we give an algorithm that exactly computes the distribution of K (i.e., qk := P(K = k), k = 0, 1, ..., n) for finite sample spaces. Finally, we give the conclusion and area for further research.
机译：在本文中，我们介绍了一种简单的强大方法，我们将独立变量λ1，...，λn替换为各种对称函数以及vieta的公式，通过事件ai的指示功能，i = 1，...，n ，即，λi= 1（ai），i = 1，...，n。计算实际发生的事件数量的随机变量k和所提出的明显同一性πni= 1（z？1（ai））≡（z≤1）k zn？k，这完全取决于k起核心作用在这篇文章中。只需选择Z（真实，复杂，随机）的不同值，并考虑各种功能，我们提供了其他简单的已知结果证明以及获得新结果。用于计算基于复杂域（Z∈C）中的IFFT的最小二乘和实体域（Z∈R）中的IFFT中的最小二乘来计算预期基本对称功能的估计算法是值得注意的。同样，我们使用牛顿的身份和一些知名的不等式来获得新的结果和不平等。然后，我们提供了一种算法，其精确计算有限样本空间的K（即，QK：= P（k = k），k = 0,1，...，n）。最后，我们提供了进一步研究的结论和地区。

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