The failure analysis of uncertain multi-degree-of-freedom (MDOF) nonlinear vibration systems with uncorrelated failure modes subjected to random excitation is examined. An earlier version of the statistical fourth-moment method is extended to vector-valued and matrix-valued functions and is developed to determine the first four moments of the system response and state function. Random variables and system derivatives are conveniently arranged into 2D matrices by means of Kronecker products. The distribution function of the system state function is approximately determined by the standard normal distribution functions using Edgeworth series technique and its failure probability is obtained. The method is based on matrix calculus. Kronecker algebra is used in the mathematical development.