The variance of the displacement, the probability densities of the displacement, velocity and amplitude and the joint probability density of the displacement and velocity of the stationary response of a Duffing oscillator to narrow-band Gaussian random excitation for a number of combinations of the parameters of the oscillator and excitation obtained from digital simulation are presented. Based on these results the stochastic jump and bifurcation of the system are examined. It is shown that for each combination of the parameters all the statistics of the stationary response are unique and independent of initial conditions. In a certain domain of the parameter space there are two more probable motions in the stationary response and jumps may occur. The jump is a transition from one more probable motion to another or vice versa. Outside of this domain the stationary response is either nearly Gaussian or like a diffused limit cycle. As the parameters cross the boundary of the domain the qualitative behavior of the stationary response changes and it is a special kind of bifurcation. It is pointed out that some previous results and explanations on this problem are incorrect.