The optimization problems with probability distributions solutions (Q2737535)

The aim of the paper is to obtain Farlie-Gumbel-Morgenstern (FGM) probability distributions for a finite number of dependent random variables by minimizing the Cramér-Pearson \(\chi\) indicator divergence, with restrictions in the form of inequalities on the covariance of the random variables. The authors obtain, for two dependent random variables, a formula for their joint distribution provided that their covariance is finitely bounded, prove that this distribution belongs to the FGM family, and decompose the joint distribution resulted formula into a FGM component and a linear component. The results for the case of two (dependent) random variables are then extended to the finite set of \(n\) random variables.NEWLINENEWLINEFor the entire collection see [Zbl 0938.00016].