The nonlinear random response of the single-degree-of-freedom system having the elasto-plastic hysteretic restoring force together with the viscous damping is presented. The Gaussian white noise is assumed as the ground acceleration. Two types of damping where the coefficient of viscousity is taken proportional to the elastic stiffness and to the instantaneous stiffness are dealt with. The analysis aims at finding the approximate solutions for the expectation and the variance of displacement, velocity, ductility factor, cumulative ductility factor and damping energy. The solutions are compared and well agree with the digital estimates in many cases.