It is pointed out that the fine-grained probability density of statistical mechanics is of interest only through coarse-grained densities—integrals over nonzero volumes of phase space. This suggests the definition of a smoothed probability density: the unsmoothed density convoluted with a kernel having a small spread aroundzero velocity. If this kernel is of Gaussian form, the smoothed density satisfies a closed and exact equation for its evolution differing from the Liouville equation by the addition of one term. This equation is applied to the simple example of a noninteracting system. We need make no assumption about the size of the system in our discussion, though if the system is large enough, the assumption that it is infinite gives the same results. Reduced distribution functions are then discussed, and a treatment of the Landau damping of electron plasma oscillations is given that is free from the usual difficulties occasioned by the breakdown of the linearization.