A Monte Carlo procedure which makes it practicable to extend the calculation of hot electron distribution functions to rarely occupied ranges of the electron state is described. The method is illustrated by a calculation of the energy distribution, for semiconductor electrons in an electric field, as a function of distance in the drift direction from the initial position. For energies comparable to the ultimate average energy, the ’’thermalization’’ of the distribution occurs close to the starting point, but with increasing energy it occurs at increasing distances.