The Maxwell-Boltzmann Distribution (MBD) for a classical ideal gas is obtained as the mean over phase space of the number of momentum components less than some value p, without making any assumptions beyond those implicit in the microcanonical ensemble. The variance of the fluctuations of actual distributions of the momentum around the average given by the MBD is shown to go to zero asN−1,whereNis the number of particles. This proof of the MBD has the advantage over the usual type of proof in that it does not employ a subdivision of phase space into cells, a trick which gives rise to certain difficulties discussed in the introduction. Moreover, it is more complete in that the problem of the fluctuations is treated at an elementary level.