This paper addresses the nonlinear random vibration of a rectangular von Karm an plate excited by uniformly distributed Gaussian white noise which is fully correlated in space. The state-space-split method and exponential polynomial closure method are jointly utilized to analyze the probabilistic solutions of the plate. The computational efficiency and numerical accuracy of the methodology for analyzing the nonlinear random vibration of the plate are verified by comparing the computational effort and numerical results with those obtained by Monte Carlo simulation and equivalent linearization, respectively. Meanwhile, the convergence of the probabilistic solution in the sense of Galerkin's approximation is examined by analyzing the plate modeled as single-degree-of-freedom and multi-degree-of-freedom systems. Some phenomena are discussed after numerically studying the behaviors of probabilistic solutions of the deflection at different locations of the plate.