The nonlinear random vibration of the simply-supported rectangular isotropic von Kármán plate with in-plane stretched edges and excited by uniformly distributed Gaussian white noise is analyzed. The equation of motion of the plate with large deflection is a nonlinear partial differential equation in space and time. The multi-degree-of-freedom nonlinear stochastic dynamical system can be formulated by applying the Galerkin's method to the nonlinear partial differential equation. The probabilistic solution of the multi-degree-of-freedom nonlinear stochastic dynamical system is governed by the Fokker-Planck-Kolmogorov equation. The state-space-split method is used to make the Fokker-Planck-Kolmogorov equation in high dimensional space reduced to the Fokker-Planck-Kolmogorov equations in 2-dimensional space. Then the exponential polynomial closure method is used to solve the reduced Fokker-Planck-Kolmogorov equations in 2-dimensional space for the probability density function of the responses of the plate with moderately large deflection.