Using a transverse electric/transverse magnetic decomposition for an angular plane wave spectrum of random electromagnetic waves and matched boundary conditions, we derive the probability density function for the energy density of the vector electric field in the presence of a semi-infinite isotropic medium. The theoretical analysis is illustrated with calculations and results for good electric conductors and for a lossless dielectric half-space. The influence of the permittivity and conductivity on the intensity, polarization state, statistical distribution, and standard deviation of the field is investigated, both for incident plus reflected fields and for refracted fields. External refraction is found to result in compression of the fluctuations of the random field. Several applications of the theory are discussed.