  Using a random-matrix approach and Monte-Carlo simulations, we generate
scattering matrices and cross sections for compound-nucleus reactions. In the
absence of direct reactions we compare the average cross sections with the
analytic solution given by the Gaussian Orthogonal Ensemble (GOE) triple
integral, and with predictions of statistical approaches such as the ones due
to Moldauer, to Hofmann, Richert, Tepel, and Weidenm\"{u}ller, and to Kawai,
Kerman, and McVoy. We find perfect agreement with the GOE triple integral and
display the limits of validity of the latter approaches. We establish a
criterion for the width of the energy-averaging interval such that the relative
difference between the ensemble-averaged and the energy-averaged scattering
matrices lies below a given bound. Direct reactions are simulated in terms of
an energy-independent background matrix. In that case, cross sections averaged
over the ensemble of Monte-Carlo simulations fully agree with results from the
Engelbrecht-Weidenm\"{u}ller transformation. The limits of other approximate
approaches are displayed.
