Abstract. The author solves the problem of finding exact lower bounds for the probability F (v) − F (u), 0<u <v <∞, where u = m − σ _μ 3√3, v = m + σ _μ 3√3, and σ _μ is a fixed dispersion in the set of distribution functions F (x) of non-negative random variables with unimodal differentiable density with mode m and two first fixed moments μ ₁, μ ₂. The case is considered where the mode coincides with the first moment: m = μ ₁. The greatest lower bound of all possible exact lower bounds for this problem is obtained and it is nearly one, namely, is equal to 0.98430.

Keywords: extremum of a linear functional, the set of unimodal distribution functions with two first fixed moments, partition of the domain of parameters.