Motivated by problems associated with the emerging theory of distributionally robust Monte Carlo simulation, this paper addresses the solution of a linear system of equations with n×n matrix A and n×1 vector b depending on an m-tuple of uncertain parameters with components entering into A and b in a rank-one manner. For such a system, the following convexity problem is considered: Determine if the second partial derivative of a solution component with respect to a specified parameter is positive over a prescribed hypercube of given radius. The main result of this paper is an extreme point solution of this problem. To this end, a factorization of the second partial derivative is provided, which plays a major role in obtaining the so-called radius of convexity. This result is then shown to be applicable within the context of a newly emerging line of research involving the use of Monte Carlo methods to compute a so-called distributionally robust expected value. This end, the expected value of solution components obtained by matrix inversion, is considered.