Abstract. The authors solve problems of finding exact lower bounds for the probability F(v) - F(u), 0 < u < v < ∞, in the set of distribution functions F(x) of nonnegative random variables with unimodal density with mode m, u < m < v, and two first fixed moments.

Keywords: extremum of a linear functional, unimodal distribution function with mode m and two first moments, partition of the domain of parameters.