We investigate the statistical properties of $C=uvu^{-1}v^{-1}$, when $u$ and
$v$ are independent random matrices, uniformly distributed with respect to the
Haar measure of the groups $U(N)$ and $O(N)$. An exact formula is derived for
the average value of power sum symmetric functions of $C$, and also for
products of the matrix elements of $C$, similar to Weingarten functions. The
density of eigenvalues of $C$ is shown to become constant in the large-$N$
limit, and the first $N^{-1}$ correction is found.
