  The paper is devoted to the study of the unconditional extremal problem for a
fractional linear integral functional defined on a set of probability
distributions. In contrast to results proved earlier, the integrands of the
integral expressions in the numerator and the denominator in the problem under
consideration depend on a real optimization parameter vector. Thus, the
optimization problem is studied on the Cartesian product of a set of
probability distributions and a set of admissible values of a real parameter
vector. Three statements on the extremum of a fractional linear integral
functional are proved. It is established that, in all the variants, the
solution of the original problem is completely determined by the extremal
properties of the test function of the linear-fractional integral functional;
this function is the ratio of the integrands of the numerator and the
denominator. Possible applications of the results obtained to problems of
optimal control of stochastic systems are described.
