On the basis of a quantum approach, the formula for the complex self-energy, and hence also that for the energy-loss cross-section, are derived for fast electrons penetrating through the surface from the interior of a solid. Both the angular and depth dependence are included in the expression. Use of a Drude-Lindhard model bulk dielectric function for describing a bulk plasmon excitation in a free-electron-like material has yielded an analytical expression for the surface dielectric function which satisfies surface sum rules. The calculated imaginary and real parts of the electron self-energy for Si have indicated, in detail, the excitation processes of the surface plasmon and the bulk plasmon for an electron passing through the surface, with the dependency on electron energy and take-off angle. This approach provides quantitatively the electron inelastic scattering cross-section in the surface region for use in surface electron spectroscopy.