We calculate the joint probability distribution of the Wigner-Smith time-de
lay matrix Q = -i (h) over bar S(-1)partial derivative S/partial derivative
epsilon and the scattering matrix S for scattering from a chaotic cavity w
ith ideal point contacts. To this end 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 La
guerre 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 o
f a quantum dot.