In this paper, a strongly nonlinear beam-impact system under both broad-banded and small-banded, Gaussian noise excitations is investigated. The response of this system is investigated both numerically, through a multi-degree-of-freedom (MDOF) model, and experimentally focusing on frequency-domain characteristics such as stochastic equivalents of harmonic and subharmonic solutions. Improved understanding of these stochastic response characteristics is obtained by comparing them to nonlinear periodic response features of the system. It will be shown that in modeling such a continuous linear system with a local nonlinearity, the linear part can be effectively reduced to a description based on several modes. Combining this reduced linear part with the local nonlinearity in a reduced nonlinear model is shown to result in an analysis model, which can be used to accurately predict the stochastic response characteristics of the original, continuous, nonlinear system. It is shown that the inclusion of more modes in the model will result in a response, which differs significantly from that of a SDOF (single-degree-of-freedom) model, giving a better correspondence with experimental results, also in the frequency range of the first mode.