Reduced velocity distributions are derived from three-dimensional measurements of the velocity distribution of electrons in the 7-500 eV range in the electron foreshock. Bump-on-tail reduced distributions are presented for the first time at the foreshock boundary consistent with Filbert and Kellogg's proposed time-of-flight mechanism for generating the electron beams. In a significant number of boundary crossings, bump-on-tail reduced distributions were found in consecutive 3 sec measurements made 9 sec apart. We conclude that, although the beams are linearly unstable to plasma waves according to the Penrose criterion, they persist on a time scale of 3-15 sec.