  We study on-site occupation number fluctuations in a system of interacting
bosons in an optical lattice. The ground-state distribution is obtained
analytically in the limiting cases of strong and weak interaction, and by means
of exact Monte Carlo simulations in the strongly correlated regime. As the
interaction is increased, the distribution evolves from Poissonian in the
non-interacting gas to a sharply peaked distribution in the Mott-insulator (MI)
regime. In the special case of large occupation numbers, we demonstrate
analytically and check numerically that there exists a wide interval of
interaction strength, in which the on-site number fluctuations remain Gaussian
and are gradually squeezed until they are of order unity near the superfluid
(SF)-MI transition. Recently, the on-site number statistics were studied
experimentally in a wide range of lattice potential depths [Phys. Rev. Lett.
\textbf{96}, 090401 (2006)]. In our simulations, we are able to directly
reproduce experimental conditions using temperature as the only free parameter.
Pronounced temperature dependence suggests that measurements of on-site atom
number fluctuations can be employed as a reliable method of thermometry in both
SF and MI regimes.
