Some general properties of the equilibrium Wigner distribution function are discussed. An invariance law under Galilean transformation is obtained for systems with no external forces. The reduced distribution functions in configurational, momentum, and phase space are studied, with emphasis on the correlations of the velocities among themselves and between them and the configurational coordinates. It is shown that up to linear terms in h2, the reduced distribution can be derived from a Hamiltonian with momentum-dependent potential.