Probability density of quantum expectation values

We consider the quantum expectation value A=〈ψ|A|ψ〉 of an observable A over the state |ψ〉. We derive the exact probability distribution of A seen as a random variable when |ψ〉 varies over the set of all pure states equipped with the Haar-induced measure. To illustrate our results we compare the exact predictions for few concrete examples with the concentration bounds obtained using Levyʼs lemma. We also comment on the relevance of the central limit theorem and finally draw some results on an alternative statistical mechanics based on the uniform measure on the energy shell.