Optimality conditions for problems with probabilistic constraints and randomizing

Optimality conditions are found for problems whose solutions are both deterministic variables and probabilistic distributions found on a set of admissible values of randomized variables. From the resultant relations follow optimality conditions for both earlier problems and new statements. A class of averaged problems is introduced regarded as problems which deal only with mean values of the determinant functions and functions of mean values of the variables. For any averaged problems the optimal distributions are shown to be concentrated in a finite number of points which are basic solutions.