THE PROBLEM OF MINIMIZING DISPERSION

Abstract

The work is related to the problem of finding the probability density of a continuous random variable in such a way that the variance or initial moments of this variable would be the smallest. The problem of minimizing the variance is solved under the condition that the probability density u x of a continuous random variable does not exceed a given function q x . The result of the work is obtaining such a density: it should be equal to the given function q x in some interval , , outside this interval the density should be zero. The numbers α and β are find from the system of nonlinear equations and are included in these equations as limits of definite integrals. By substituting the found function in the extreme condition, the smallest variance value for any continuous random variable is obtained. The problem of finding the probability density of a random variable is also considered under the conditions x u x dx u x dx u x q k              min, 1, 0 , where q and k are given positive numbers. The described problems belong to the class of Lyapunov extremal problems containing integrals. When solving them, the minimum principle and the method of Lagrange multipliers were applied. As a consequence of the Weierstrass theorem, it follows that the found stationary points are the points of the smallest values of the considered functionals. Substituting the solution of the problem into an extreme condition, an inequality for the initial moments of any continuous random variable, which is bounded from above by a positive constant q is obtained. It is emphasized that the determination of parameters as a result of minimization will allow to determine more accurate estimates in other values and to achieve an effective level of parameter optimization. The results of solving mathematical problems are given. An exemplary application of the given problem is the implementation of adaptive forecasting and statistical identification of nonlinear dynamic models of physical and economic processes. The scientific novelty of the obtained solution lies in the fact that, for the first time, a system was obtained in which the mathematical expectation of an extreme random variable is located in the middle of the segment , , due to which, by substituting the found function in an extreme condition, the smallest variance value for any continuous random variable is obtained.