Optimization Problems in Forming a Mixture of Distribution Functions of Operating Times to Failure of Elements of Technical Systems

The distribution densities of the laws of distribution of random operating times known in the theory of reliability, for example, exponential, Weibull–Gnedenko, Erlang, normal, Maxwell, and many others are not more than unimodal and not more than two-parameter. This paper discusses formulation of problems of determining the share of each distribution function in a mixture (distribution functions are given) for which a random variable specified by this mixture of distributions has the least variance under a certain expectation or the largest expectation under a certain variance. The problems are formulated as the well-known Markowitz problems on selecting a block of securities under the assumption that the mean represents the return and the variance represents the risk. Solutions to the problem of minimizing the variance for mixtures of two and three distributions for a certain expectation are presented.