The multi-degree-of-freedom system is formulated from the typical problem of the stretched nonlinear Euler-Bernoulli beam excited by filtered white noise. The probabilistic solution of the multi-degree-of-freedom nonlinear stochastic dynamic systems similar to that formulated from Euler-Bernoulli beam and excited by filtered white noise is obtained by the state-space-split method and exponential polynomial closure method. The way for selecting the sub-state vectors in the solution procedure with the state-space-split method is given for the discussed problem. The solution procedure with the state-space-split method is presented for the system excited by filtered white noise. Numerical results are presented. The results obtained with the state-space-split method and exponential polynomial closure method are compared with those obtained by Monte Carlo simulation and equivalent linearization method to verify the effectiveness of the state-space-split method and exponential polynomial closure method in analyzing the probabilistic solutions of the multi-degree-of-freedom nonlinear stochastic dynamic systems similar to that formulated from the stretched nonlinear Euler-Bernoulli beam.