  We consider the statistics of time delay in a chaotic cavity having $M$ open
channels, in the absence of time-reversal invariance. In the random matrix
theory approach, we compute the average value of polynomial functions of the
time delay matrix $Q=-i\hbar S^\dag dS/dE$, where $S$ is the scattering matrix.
Our results do not assume $M$ to be large. In a companion paper, we develop a
semiclassical approximation to $S$-matrix correlation functions, from which the
statistics of $Q$ can also be derived. Together, these papers contribute to
establishing the conjectured equivalence between the random matrix and the
semiclassical approaches.
