Greatest Lower Bound of System Failure Probability on a Special Time Interval Under Incomplete Information About the Distribution Function of the Time to Failure of the System

The author solves the problem of finding greatest lower bounds for the probability F (??) – F (u),0 < u <, ?? < ∞, where \( u= m-{\upsigma}_{\mu}3\sqrt{3},\kern0.5em \upupsilon = m+{\upsigma}_{\mu}3\sqrt{3},\kern0.5em \mathrm{and}\kern0.5em {\upsigma}_{\mu} \) 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 and μ ₂. The case is considered where the mode coincides with the first moment: m = μ ₁. The greatest lower bound of all possible greatest lower bounds for this problem is obtained and it is nearly one, namely, 0.98430.