For reduction of seismic response of the mechanical system, friction mechanism is utilized in base isolation systems. In general, responses of the mechanical systems decrease with the increase of friction coefficient. However, in acceleration responses, the optimum friction coefficient which gives the minimum response of the mechanical system exists in some conditions of experiments. This phenomenon is examined using random vibration theory. The mechanical system including base isolation system is modeled as a single-degree-of-freedom system with friction element. As seismic excitations, nonstationary filtered white noises are used. The mean square value of absolute acceleration and relative displacement of the mechanical system to excitation are obtained using equivalent linearization method. The maximum response is defined as the maximum value of the standard deviation of the response. The optimum friction coefficient exists when excitation is small. For characteristics of input excitation, band width and duration of excitation are considered. The effect of input excitation characteristic on the optimum friction force is examined.