Using the idea of random function, the standard orthogonal random variables in the spectral representation method is expressed as the orthogonal function form of basic random variables. Through constructing two different forms of stochastic orthogonal trigonometric functions, non-Gaussian orthogonal random variables and Gaussian independent random variables can be represented by the orthogonal trigonometric functions of the basic random variables. Compared to existing spectral representations of stochastic process, one or two basic random variables can capture the second-order statistics of the original stochastic process, and sample functions with assigned probability on the non-Gaussian stationary process or the Gaussian stationary process can be directly generated by the power spectral density function. Finally, combining the power spectral density function of stationary ground motion acceleration process, a numerical example is given to illustrate the effectiveness of the hybrid random function and spectral representation approach.