Abstract: The classical and quantum statistical mechanics discussed in detail are those of hypotheses which possess axiomatic properties. Based on the hypotheses Maxwell-Boltzmann, Bose-einstein, and Fermi-Dirac statistical distributions are derived and proved concisely and rigorously. Also the principle of quantum grand canonical ensemable is analysed. And it is indicated that the non-identity and identity of bodies essentially result in the diversity between calssical and quanrum statistical mechanics: the forrner is related to the total body number and the latter is not.