One of the key issues in quantum information theory related problems concerns
with that of distinguishability of quantum states. In this context, Bures
distance serves as one of the foremost choices among various distance measures.
It also relates to fidelity, which is another quantity of immense importance in
quantum information theory. In this work, we derive exact results for the
average fidelity and variance of the squared Bures distance between a fixed
density matrix and a random density matrix, and also between two independent
random density matrices. These results supplement the recently obtained results
for the mean root fidelity and mean of squared Bures distance [Phys. Rev. A
104, 022438 (2021)]. The availability of both mean and variance also enables us
to provide a gamma-distribution-based approximation for the probability density
of the squared Bures distance. The analytical results are corroborated using
Monte Carlo simulations. Furthermore, we compare our analytical results with
the mean and variance of the squared Bures distance between reduced density
matrices generated using coupled kicked tops, and a correlated spin chain
system in a random magnetic field. In both cases, we find good agreement.