The nonzero-mean probability density function (PDF) solutions of nonlinear stochastic oscillators under the excitation of Poisson impulse process are obtained with exponential-polynomial closure (EPC) method. The excitations are assumed to be external Poisson impulse process and parametric Poisson impulse process on displacement. The PDF of the oscillator response is governed by the generalized Fokker-Planck-Kolmogorov (FPK) equation, which is solved with the EPC method. The nonlinear oscillator with external and parametric excitation on displacement is analyzed when the mean of oscillator response is nonzero. Different levels of oscillator nonlinearity and nonzero means of the impulse amplitude are considered in the analysis. The analytical results show that the PDF solutions given by the EPC method are in good agreement with the simulated results when the complete sixth-degree polynomial of state variables is taken in the EPC procedure. The good agreement is also observed in the tail regions of the PDF solutions. The numerical analysis further shows that the PDF of displacement is not symmetrical about the mean of displacement.