Improved Monte Carlo models for the concentration and velocity distribution for hydrogen in the terrestrial exosphere have been formulated for minimum, medium, and maximum solar cycle conditions. Both the plasmaspheric source of hot hydrogen and the classical exobase source have been included, along with solar radiation pressure and photoionization. At solar minimum the hydrogen from charge exchange of hot ions in the plasmasphere exceeds that from the exobase source not only for escape but for the population at geocentric distances greater than two earth radii (RE). At about 2 RE the equivalent temperature is about 50% greater than that of the exobase, a situation similar to the ''two-temperature'' exosphere observed and calculated for Venus. At solar maximum the exobase source dominates. The concentration at about 2 RE geocentric and above varies little from solar maximum to solar minimum, although the exobase concentration increases by about a factor of 10. The diurnal variation at the exobase is a factor of 2.1 at solar maximum and 2.7 at solar minimum at low latitudes. This variation is mostly smoothed out at 2 RE, but above 10 RE a nighttime enhancement by about a factor of 2 develops, constituting a slight ''geotail'' of the type found earlier by observation and by calculations of the effects of radiation pressure. The non-Maxwellian velocity distributions show ''flattened'' tops compared to Maxwellian distributions for components in the transverse directions between 2 RE and 8RE which appear to be due to the importance there of satellite particles with high apogees and the depletion of satellite particles with low perigees.