An orthogonal expansion method for earthquake ground motion was introduced in the first part of the paper. In the method, seismic acceleration process is represented as a linear combination of deterministic functions modulated by 10 uncorrelated random variables. In the second part of the paper, the recently developed probability density evolution method (PDEM) is employed to study linear random response of structures subjected to the external excitations. In the PDEM, a completely uncoupled one-dimensional governing partial differential equation, the generalized density evolution equation, is derived first with regard to evolutionary probability density function of the stochastic response for nonlinear structures. The solution of this equation can put out the instantaneous probability density function. So it is natural to combine the PDEM and the orthogonal expansion of seismic ground motion to study the linear random earthquake response. Finally, combining an example of a linear frame structure subjected to non-stationary ground motions, this paper validate the proposed approach and expounds the application of this method.