  Vertical loads acting on the surface of a half-space made of discrete and
elastic particles are supported by a network of force chains that changes with
the specific realization of the packing. These force chains can be transformed
into equivalent stress fields, but the obtained values are usually different to
those expected from the solution of the corresponding boundary value problem.
In this research the relationship between discrete and continuum approaches to
Boussinesq-like problems is explored in the light of classical statistical
mechanics. In principal directions, the anticipated statistical distributions
of the extensive stress (\textit{i.e.} the product of the stress by the volume)
are exponential distributions for normal components and Laplace distributions
for shear components. The parameters scaling these distributions can be
obtained from the solutions provided by continuum approaches in most of the
cases. This has been validated through massive numerical simulation with the
discrete element method. These results could be of interest in highly
fragmented, faulted or heterogeneous media or for small length scales.
