The response of non-linear, dynamic systems to stochastic excitation exhibits many interesting characteristics. In thispaper, a strongly non-linear beam-impact system under both broad- and small-banded, Gaussian noise excitations isinvestigated. The response of this system is investigated both numerically, through a multi-degree-of-freedom model, andexperimentally focusing on frequency-domain phenomena such as stochastic equivalents of harmonic and subharmonicsolutions. An improved understanding of these stochastic response characteristics is obtained by comparing these tonon-linear periodic response features of the system. It will be shown that in modelling such a continuous, linear systemwith a local non-linearity, the linear part can be effectively reduced to a description based on several modes. Combining thisreduced, linear part with the local non-linearity in a reduced, non-linear model is shown to result in a non-linear model,which can be used to accurately predict the stochastic response characteristics of the original, continuous, non-linearsystem. It is shown that including more modes to the model causes its response to differ significantly from that of asingle-degree-of-freedom model and show a better correspondence with experimental results, also in the frequency rangeof the first mode.