  The goal of this paper is to calculate exactly the average of
uncertainty-product of two bounded observables and to establish its typicality
over the whole set of finite dimensional quantum pure states. Here we use the
uniform ensembles of pure and isospectral states as well as the states
distributed uniformly according to the measure induced by the Hilbert-Schmidt
norm. Firstly, we investigate the average uncertainty of an observable over
isospectral density matrices. By letting the isospectral density matrices be of
rank-one, we get the average uncertainty of an observable restricted to pure
quantum states. These results can help us check how large the gap is between
the uncertainty-product and any obtained lower bounds about the
uncertainty-product. Although our method in the present paper cannot give a
tighter lower bound of uncertainty-product for bounded observables, it can help
us drop any one that is not tighter than the known one substantially.
