  We explore how the expectation values $\langle\psi |A| \psi\rangle$ of a
largely arbitrary observable $A$ are distributed when normalized vectors
$|\psi\rangle$ are randomly sampled from a high dimensional Hilbert space. Our
analytical results predict that the distribution exhibits a very narrow peak of
approximately Gaussian shape, while the tails significantly deviate from a
Gaussian behavior. In the important special case that the eigenvalues of $A$
satisfy Wigner's semicircle law, the expectation value distribution for
asymptotically large dimensions is explicitly obtained in terms of a large
deviation function, which exhibits two symmetric non-analyticities akin to
critical points in thermodynamics.
