  We consider wave propagation in a complex structure coupled to a finite
number $N$ of scattering channels, such as chaotic cavities or quantum dots
with external leads. Temporal aspects of the scattering process are analysed
through the concept of time delays, related to the energy (or frequency)
derivative of the scattering matrix $\mathcal{S}$. We develop a random matrix
approach to study the statistical properties of the symmetrised Wigner-Smith
time-delay matrix
$\mathcal{Q}_s=-\mathrm{i}\hbar\,\mathcal{S}^{-1/2}\big(\partial_\varepsilon\mathcal{S}\big)\,\mathcal{S}^{-1/2}$,
and obtain the joint distribution of $\mathcal{S}$ and $\mathcal{Q}_s$ for the
system with non-ideal contacts, characterised by a finite transmission
probability (per channel) $0<T\leq1$. We derive two representations of the
distribution of $\mathcal{Q}_s$ in terms of matrix integrals specified by the
Dyson symmetry index $\beta=1,2,4$ (the general case of unequally coupled
channels is also discussed). We apply this to the Wigner time delay
$\tau_\mathrm{W}=(1/N)\,\mathrm{tr}\big\{\mathcal{Q}_s\big\}$, which is an
important quantity providing the density of states of the open system. Using
the obtained results, we determine the distribution
$\mathscr{P}_{N,\beta}(\tau)$ of the Wigner time delay in the weak coupling
limit $NT\ll1$ and identify three different asymptotic regimes.
