  We consider the quantum expectation value \mathcal{A}=\<\psi|A|\psi\> of an
observable A over the state |\psi\> . We derive the exact probability
distribution of \mathcal{A} seen as a random variable when |\psi\> varies over
the set of all pure states equipped with the Haar-induced measure. The
probability density is obtained with elementary means by computing its
characteristic function, both for non-degenerate and degenerate observables. To
illustrate our results we compare the exact predictions for few concrete
examples with the concentration bounds obtained using Levy's lemma. Finally we
comment on the relevance of the central limit theorem and draw some results on
an alternative statistical mechanics based on the uniform measure on the energy
shell.
