We study the fluctuations of the probability density [ital P]([ital r],[ital t]) of diffusing particles to be at distance [ital r] at time [ital t] in the presence of random potentials, represented by random transition rates. We find an exact relation which expresses all the moments of [ital P]([ital r],[ital t]) in terms of its first moment, for both quenched and annealed disorder and for any dimension. From this relation follows that anomalous diffusion implies nontrivial behavior of the moments of [ital P]([ital r],[ital t]), such as an exponential divergence of the relative fluctuations for large [ital r].