We examine the steady state properties of binary systems of driven inelastic hard spheres. The spheres, which move under the influence of gravity, are contained in a vertical cylinder with a vibrating base. We computed the trajectories of the spheres using an event-driven molecular dynamics algorithm. In the first part of the study, we chose simulation parameters that match those of experiments published by Wildman and Parker. Various properties computed from the simulation including the density profile, granular temperature, and circulation pattern are in good qualitative agreement with the experiments. We then studied the effect of varying the mass ratio and the size ratio independently while holding the other parameters constant. The mass and size ratio are shown to affect the distribution of the energy. The changes in the energy distributions affect the packing fraction and temperature of each component. The temperature of the heavier component has a nonlinear dependence on the mass of the lighter component, while the temperature of the lighter component is approximately proportional to its mass. The temperature of both components is inversely dependent on the size of the smaller component.