We simulate the spatial and spectral distributions of plasmaspheric hiss using a technique that involves extensive ray tracing. The rays are injected in the equatorial chorus source region outside the plasmasphere, are power weighted as a function of L-shell, frequency, and wave normal angle, so as to represent the chorus source distribution, and are propagated throughout the simulation domain until the power in each ray is effectively extinguished due to Landau damping. By setting up a large number of virtual observatories, the rays passing each observation location are counted, and a distribution is constructed. Our simulated plasmaspheric hiss spectrum reproduces the main observed features, including the lower and upper frequency cutoffs, the behavior of the bandwidth as a function of L-shell, the spatial extent, and even the two-zone structure of hiss, although the intensity is lower than observed. The wave normal distribution shows that at high latitudes, the wave normals are predominantly oblique, but near the equator, the wave normal distribution can be either predominantly field-aligned (lower L shells), or be bimodal, having a maximum in the field-aligned direction, and another maximum at very oblique angles, comprised of those rays that have broken out of their cyclical trajectories. This distribution of wave normals seems to reconcile the apparently contradictory observations that have been reported previously.