  We study the set of random matrix product states (RMPS) introduced in
arXiv:0908.3877 as a tool to explore foundational aspects of quantum
statistical mechanics. In the present work, we provide an accurate numerical
and analytical investigation of the properties of RMPS. We calculate the
average state of the ensemble in the non-homogeneous case, and numerically
check the validity of this result. We also suggest using RMPS as a tool to
approximate properties of general quantum random states. The numerical
simulations presented here support the accuracy and efficiency of this
approximation. These results suggest that any generalized canonical state can
be approximated with high probability by the reduced density matrix of a random
MPS, if the average MPS coincide with the associated microcanonical ensemble.
