  Embedded random matrix ensembles are generic models for describing
statistical properties of finite isolated interacting quantum many-particle
systems. For the simplest spinless systems, with say $m$ particles in $N$
single particle states and interacting via $k$-body interactions, we have
EGUE($k$) and the embedding algebra is $U(N)$. A finite quantum system, induced
by a transition operator, makes transitions from its states to the states of
the same system or to those of another system. Examples are electromagnetic
transitions (same initial and final systems), nuclear beta and double beta
decay (different initial and final systems), particle addition to/removal from
a given system and so on. Towards developing a complete statistical theory for
transition strength densities, we have derived formulas for lower order
bivariate moments of the strength densities generated by a variety of
transition operators. For a spinless fermion system, using EGUE($k$)
representation for Hamiltonian and an independent EGUE($t$) representation for
transition operator, finite-$N$ formulas for moments up to order four are
derived, for the first time, for the transition strength densities. Formulas
for the moments up to order four are also derived for systems with two types of
spinless fermions and a transition operator similar to beta decay and
neutrinoless beta decay operators. Moments formulas are also derived for
transition operator that removes $k_0$ number of particles from $m$ fermion
system. Numerical results obtained using the exact formulas for two-body
($k=2$) Hamiltonians (in some examples for $k=3,4$) and the asymptotic formulas
clearly establish that in general the smoothed form of the bivariate transition
strength densities take bivariate Gaussian form for isolated finite quantum
systems. Extensions of these results to bosonic systems and EGUE ensembles with
further symmetries are discussed.
