  We calculate the joint probability distribution of the Wigner-Smith
time-delay matrix $Q=-i\hbar S^{-1} \partial S/\partial \epsilon$ and the
scattering matrix $S$ for scattering from a chaotic cavity with ideal point
contacts. Hereto we prove a conjecture by Wigner about the unitary invariance
property of the distribution functional $P[S(\epsilon)]$ of energy dependent
scattering matrices $S(\epsilon)$. The distribution of the inverse of the
eigenvalues $\tau_1,...,\tau_N$ of $Q$ is found to be the Laguerre ensemble
from random-matrix theory. The eigenvalue density $\rho(\tau)$ is computed
using the method of orthogonal polynomials. This general theory has
applications to the thermopower, magnetoconductance, and capacitance of a
quantum dot.
