  A model is presented here that is capable of simultaneously describing the
luminosity function and the underlying joint population distribution of the
prompt spectral and temporal parameters of Gamma-Ray Bursts (GRBs), subject to
the detection threshold of gamma-ray instruments--in particular, BATSE and
Fermi. Based on 2130 GRB prompt emission data in the BATSE catalog, I show that
the population properties of the two classes of GRBs -- Long and Short
durations -- bear striking similarities in the 4-dimensional space of prompt
parameters: peak luminosity (Liso), total isotropic emission (Eiso),
time-integrated spectral peak energy (Ep,z) and the prompt duration (T90,z).
The two well-known Amati (Eiso-Ep,z) and Yonetoku (Liso-Ep,z) relations are
shown to be highly affected by selection effects, undermining the legitimacy of
their frequent uses in the studies of Dark Energy's equation of state and
different cosmological models. In particular, I show that the slope of the
Amati relation is likely in the range 0.2-0.3, corresponding to a Pearson's
correlation strength of rho=0.58\pm0.04. This predicted slope is significantly
(>16sigma) less than the value currently perceived by the GRB community
(~0.56). I argue that similar Eiso-Ep,z and Liso-Ep,z relations with
approximately same strength and significance should exist in the population of
Short GRBs. Also predicted by the model is the strong positive correlation of
the prompt emission duration (e.g., T90,z) with Eiso and Liso in both classes
of Short and Long GRBs.
