  For finite quantum many-particle systems, a given 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 (then the initial and final systems are same), nuclear beta and
double beta decay (then the initial and final systems are different), particle
addition to or removal from a given system and so on. Working towards
developing a complete statistical theory for transition strength densities
(transition strengths multiplied by the density of states at the initial and
final energies), we have started a program to derive formulas for the lower
order bivariate moments of the strength densities generated by a variety of
transition operators. In this paper results are presented for a transition
operator that removes $k_0$ number of particle by considering $m$ spinless
fermions in $N$ single particle states. The Hamiltonian that is $k$-body is
represented by EGUE($k$) [embedded Gaussian unitary ensemble of $k$-body
interactions] and similarly the transition operator by an appropriate
independent EGUE. Employing the embedding $U(N)$ algebra, finite-$N$ formulas
for moments up to order four are derived and they show that in general the
smoothed (with respect to energy) bivariate transition strength densities take
bivariate Gaussian form. Extension of these results to particle addition
operator and beta decay type operators are discussed.
