Probability Problems and Estimation Algorithms Associated with Symmetric Functions

Abstract

In this chapter, a simple powerful methodology is presented where we replace the independent variables \(\lambda_{1}, \ldots, \lambda_{n}\)  in multiple symmetric functions as well as in Vieta’s formulas by the indication functions of the events \(A_{i}, i=1, \ldots, n \text {, i.e., } \lambda_{i}=1\left(A_{i}\right), i=1, \ldots, n\) Both the random variable K that counts the number of events which actually occurred and the proposed identity \(\prod_{i=1}^{n}\left(z-1\left(A_{i}\right)\right) \equiv\) (z-1)^{K} z^{n-K} that solely depends on K play a major role in this chapter Just by selecting multiple values for z (real, complex, and random) and taking expectations of the different functions we provide other simple proofs of known findings as well as get new results. The estimated algorithms for computing the expected elementary symmetric functions via least squares based on IFFT in the complex domain \((z \in \mathbb{C})\) and least squares or linear programming in the real domain \((z \in \mathbb{R})\) are noteworthy. we use Newton’s identities and some popular inequalities to get new outcome and inequalities. we express an algorithm that exactly computes the distribution of K (i.e., qk \(:=\mathbb{P}(K=k), k=0,1, \ldots, n)\) for finite sample spaces. Finally, we provide a conclusion and areas for future research.