This paper reports on our study of the edge of the 2/5 fractional quantum Hall state, which is more complicated than the edge of the 1/3 state because of the presence of edge sectors corresponding to different partitions of composite fermions in the lowest two Lambda levels. The addition of an electron at the edge is a nonperturbative process and it is not a priori obvious in what manner the added electron distributes itself over these sectors. We show, from a microscopic calculation, that when an electron is added at the edge of the ground state in the [N-1, N-2] sector, where N-1 and N-2 are the numbers of composite fermions in the lowest two Lambda levels, the resulting state lies in either [N-1 + 1, N-2] or [N-1, N-2 + 1] sectors; adding an electron at the edge is thus equivalent to adding a composite fermion at the edge. The coupling to other sectors of the form [N-1 + 1 + k, N-2 - k], k integer, is negligible in the asymptotically low-energy limit. This study also allows a detailed comparison with the two-boson model of the 2/5 edge. We compute the spectral weights and find that while the individual spectral weights are complicated and nonuniversal, their sum is consistent with an effective two-boson description of the 2/5 edge.