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

Solution of the Unconditional Extremal Problem for a Linear-Fractional Integral Functional Depending on the Parameter