The probability density function (PDF) solution of the response is formulated for nonlinear systems under discrete Poisson impulse excitation. The PDF solution is gov-erned by the Kolmogorov–Feller (KF) equation, which is approximately solved by the exponential–polynomial closure (EPC) method. A Duffing oscillator is further investi-gated in the case of either Gaussian or non-Gaussian distributed amplitude of Poisson impulse to show the effectiveness of the EPC method in these cases. The numerical anal-ysis shows that the EPC method with the polynomial order being 6 presents a good result compared with the simulated result, even in the tails of the PDF of the oscillator response.