  We calculate the joint distribution $P(S,Q)$ of the scattering matrix $S$ and
time-delay matrix $Q=-i\hbar S^\dagger dS/dE$ of a chaotic quantum dot coupled
by point contacts to metal electrodes. While $S$ and $Q$ are statistically
independent for ballistic coupling, they become correlated for tunnel coupling.
We relate the ensemble averages of $Q$ and $S$ and thereby obtain the average
density of states at the Fermi level. We apply this to a calculation of the
effect of a tunnel barrier on the Majorana resonance in a topological
superconductor. We find that the presence of a Majorana bound state is hidden
in the density of states and in the thermal conductance if even a single
scattering channel has unit tunnel probability. The electrical conductance
remains sensitive to the appearance of a Majorana bound state, and we calculate
the variation of the average conductance through a topological phase
transition.
