Study of power series distributions with specified covariances

This paper presents a study of power series distributions (PSD) with prescribed covariance characteristics. Such distributions constitute a fundamental class in probability theory and mathematical statistics, as they generalize a wide range of well-known discrete distributions and enable the description of various stochastic phenomena with a predetermined variance structure. The aim of the research is to develop analytical methods for constructing power series distributions with given covariances and to establish the conditions under which a particular function can serve as the covariance of a certain PSD. The paper derives a first-order differential equation for the generating function of the distribution, which determines the relationship between its parameters and the form of the covariance function. It is shown that the choice of an analytical or polynomial covariance completely specifies the structure of the corresponding generating function. The analysis made it possible to construct new families of PSDs that generalize the classical Bernoulli, Poisson, geometric, and other distributions while preserving a given covariance structure. The proposed approach is based on the analytical relationship between the generating function and the covariance function, providing a framework for constructing stochastic models with predefined dispersion properties. The results obtained expand the theoretical framework for describing discrete distributions and open up opportunities for practical applications in statistical estimation, modeling of complex systems, financial processes, machine learning where it is crucial to control the dependence between the mean and the variation. Further research may focus on constructing continuous analogues of such distributions, studying their limiting properties, and applying them to problems of regression and Bayesian analysis.