  Given a uniform ensemble of quantum density matrices $\rho$, it is useful to
calculate the mean value over this ensemble of a product of entries of $\rho$.
We show how to calculate such moments in this paper. The answer involves well
known results from Group Representation Theory and Random Matrix Theory. This
quantum problem has a well known classical counterpart: given a uniform
ensemble of probability distributions $P=(P_1, P_2, ..., P_N)$ where the $P_j$
are non-negative reals that sum to one, calculate the mean value over this
probability simplex of products of $P$ components. The answer to the classical
problem follows from an integral formula due to Dirichlet.
