The stationary probability density function (PDF) solution of random oscillators with correlated additive and multiplicative Gaussian excitations is investigated in this paper. The correlation between additive and multiplicative Gaussian excitations is taken into account. As a result, the generalized Fokker-Planck-Kolmogorov (FPK) equation is expressed with the independent part and the correlated part, which can be solved by the exponential-polynomial closure (EPC) method. The linear and nonlinear oscillators under correlated additive and multiplicative Gaussian white noise excitations are investigated. Two cases of different correlated additive and multiplicative excitations are considered. Compared with the results in the case of independent external and parametric excitations, unsymmetrical PDFs and nonzero means of system responses can be obtained.