Based on the generalized probabilistic finite element method, this paper presents anapproximate solution technique for general multi-degree-of-freedom nonlinear random vibrationsystems with random parameters. The fourth-moment technique, maximum entropy theory andincomplete probability information theory are employed to systematically develop a reliabilityanalysis method for dynamic random structural systems with correlation failure modes underunavailable joint probability density functions of basic random variables. The first passage problemof multi-degree-of-freedom nonlinear random vibration systems is solved.