A detailed mathematical analysis on the q = 1/2 non-extensive maximum entropy
distribution of Tsallis' is undertaken. The analysis is based upon the splitting of such a
distribution into two orthogonal components. One of the components corresponds to the
minimum norm solution of the problem posed by the fulfillment of the a priori conditions
on the given expectation values. The remaining component takes care of the normalization
constraint and is the projection of a constant onto the Null space of the "expectation-values-transformation"