AbstractSolutions of the moment problem are sought in the class of density functions that possess certain conditions of continuity and differentiability. One can accomplish this by the methods of linear optimization if the nodal quantities, for instance, the ordinates and derivatives are retained as independent variables and if the density function is approximated over the subintervals by an interpolation process, such as the cubic spline functions. Computations indicate that the procedure yields bounds on the probabilities that are sharper than the ones due to the classical methods.