  For the embedded Gaussian orthogonal ensemble (EGOE) of random matrices, the
strength sums generated by a transition operator acting on an eigenstate vary
with the excitation energy as the ratio of two Gaussians. This general result
is compared to exact shell model calculations, with realistic interactions, of
spherical orbit occupancies and Gamow-Teller strength sums in some $(ds)$ and
$(fp)$ shell examples. In order to confirm that EGOE operates in the chaotic
domain of the shell model spectrum, calculations are carried out using two
different interpolating hamiltonians generating order-chaos transitions. Good
agreement is obtained in the chaotic domain of the spectrum, and strong
deviations are observed as nuclear motion approaches a regular regime
(transition strength sums appear to follow the Dyson's $\Delta_3$ statistic).
More importantly, they shed new light on the newly emerging understanding that
in the chaotic domain of isolated finite interacting many particle systems
smoothed densities (they include strength functions) define the statistical
description of these systems and these densities follow from embedded random
matrix ensembles; some EGOE calculations to this end are presented.
