  Recently proposed correlations between the energetics of Gamma-Ray Bursts
(GRB) and their spectral properties, namely the peak energy of their prompt
emission, are found to account for the observed fluence distribution of all
`bright' BATSE GRB. Furthermore for an intrinsic GRB peak energy distribution
extending toward lower energies with respect to that characterizing bright GRB,
such correlations allow to reproduce the fluence distribution of the whole
BATSE long GRB population. We discuss the constraints that such analysis
imposes on the shape of the peak energy distribution, the opening angle
distribution and the tightness of such correlations.
