  Ground state properties of a fermionic Coulomb gas are calculated using the
fixed-node diffusion Monte Carlo method. The validity of the composite boson
description is tested for different densities. We extract the exciton-exciton
$s$-wave scattering length by solving the four-body problem in a harmonic trap
and mapping the energy to that of two trapped bosons. The equation of state is
consistent with the Bogoliubov theory for composite bosons interacting with the
obtained $s$-wave scattering length. The perturbative expansion at low density
has contributions physically coming from (a) exciton binding energy, (b)
mean-field Gross-Pitaevskii interaction between excitons, (c) quantum depletion
of the excitonic condensate (Lee-Huang-Yang terms for composite bosons). In
addition, for low densities we find a good agreement with the Bogoliubov
bosonic theory for the condensate fraction of excitons. The equation of state
in the opposite limit of large density is found to be well described by the
perturbative theory including (a) mixture of two ideal Fermi gases (b) exchange
energy. We find that for low densities both energetic and coherent properties
are correctly described by the picture of composite bosons (excitons).
