The energy distribution functions of nonthermal thermospheric hydrogen atoms are calculated for electron and proton precipitation in the Jovian aurora. A numerical model taking into account the production, elastic and inelastic relaxation and transport processes for hot H atoms is developed. This model is based on a Monte Carlo solution of the nonlinear Boltzmann equation for hot H atoms produced by electron and proton impact on H and H2 and exothermic chemical reactions. The distribution functions show a much higher energy tail for proton than electron precipitation. It is shown that the steady state flux of hot atoms (E ≥ 2 eV) is essentially isotropic. The peak and column hot H densities are about 3 × 105 cm−3 and 1 × 1014 cm−3 for a 100 erg cm−2 s−1 precipitation combining hard (22 keV) and soft (0.2 keV) electrons mixed with a 10 erg cm−2 s−1 flux of soft (0.3 keV) protons. These column densities, coupled with the wide range of hot H atom energies, may play an important role in the formation of the Lyman α line profile. Multiple scattering in the wings of the Ly α line by the fast H atoms is shown to partly account for the broad Ly α profile observed in the Jovian aurora with the Hubble space telescope.