  The full counting statistics of a molecular level weakly interacting with a
local phonon mode is derived. We find an analytic formula that gives the
behavior of arbitrary irreducible moments of the distribution upon phonon
excitation. The underlying competition between quasi-elastic and inelastic
processes results in the formation of domains in parameter space characterized
by a given sign in the jump of the irreducible moments. In the limit of perfect
transmission, the corresponding distribution is distorted from Gaussian
statistics for electrons to Poissonian transfer of holes above the inelastic
threshold.
