The stationary probability density function (PDF) solution of the stochastic response of nonlinear oscillators is investigated in this paper. The external excitation is assumed to be a combination of Gaussian and Poisson white noises. The PDF solution is governed by the generalized Kolmogorov equation which is solved by the exponential-polynomial closure (EPC) method. In order to evaluate the effectiveness of the EPC method, different nonlinear oscillators are considered in numerical analysis. Nonlinearity exists either in displacement or in velocity for these nonlinear oscillators. The impulse arrival rate, mono-modal PDF and bi-modal PDF are also considered in this study. Compared to the PDF given by Monte Carlo simulation, the EPC method presents good agreement with the simulated result, which can also be observed in the tail region of the PDF solution.