The probabilistic solutions of elastic stretched beams are studied when the beam is discretized by finite difference scheme and excited by Gaussian white noise which is fully correlated in space. The nonlinear multi-degree-of-freedom system about the random vibration of stretched beam is formulated by finite difference scheme first. Then the relevant Fokker-Planck-Kolmogorov equation is solved for the probabilistic solutions of the system by the state-space-split and exponential polynomial closure method. Monte Carlo simulation method and equivalent linearization method are also adopted to analyze the probabilistic solutions of the system responses, respectively. Numerical results obtained with the three methods are presented and compared to show the computational efficiency and numerical accuracy of solving the Fokker-Planck-Kolmogorov equation by the state-space-split and exponential polynomial closure method in analyzing the probabilistic solutions of the beams discretized by finite difference scheme and excited by Gaussian white noise. The techniques of using the state-space-split procedure for dimension reduction of the beam systems are discussed through the given beam systems with different space distributions of excitations.