The Langevin system subjected to non-Gaussian noise has been discussed, by using the second-order moment approach with two kinds of models for generating the noise. We have derived the effective differential equation (DE) for a variable $x$, from which the stationary probability distribution $P(x)$ has been calculated with the use of the Fokker-Planck equation. The result of $P(x)$ calculated by the moment method is compared to several expressions obtained by different methods such as the universal colored noise approximation (UCNA) [Jung and Hänggi, Phys. Rev. A {\bf 35}, 4464 (1987)] and the functional-integral method. It has been shown that our $P(x)$ is in good agreement with that of direct simulations (DSs). We have also discussed dynamical properties of the model with an external input, solving DEs in the moment method.

22 pages, 6 figures; accepted in Physica A with augmented discussions