This paper deals with the stationary probability density function (PDF) solution of the dynamic response for nonlinear oscillators excited by Poisson impulses. The PDF solution is governed by the generalized Fokker-Planck-Kolmogorov (FPK) equation. The exponential-polynomial closure (EPC) method is used to obtain an approximate solution to the generalized FPK equation. In order to evaluate the effectiveness of the EPC method in the case of Poisson excitations, Duffing oscillators, van der Pol oscillators, and an additional nonlinear oscillator excited by external and parametric Poisson impulses with different levels of nonlinearity in stiffness and damping are studied. Numerical results show good agreement with the probability distribution obtained with Monte Carlo simulation including the tail regions, which is of significance for reliability analysis.