  We use a new method of analysis to determine parameters of cosmological
gamma-ray bursts (GRBs), assuming that their distribution follows the
star-formation history of the universe. Spectral evolution is calculated from
an external shock model for fireball/blast wave evolution, and used to evaluate
the measured peak flux, duration, and $\nu F_\nu$ peak photon energy for a GRB
source occuring at a given redshift and with given values of total energy,
baryon-loading and environmental parameters. We then fit model distributions of
GRB sources to the observed peak flux, duration and $\nu F_\nu$ peak photon
energy distributions. We find that the observed width of the $E_p$ and duration
distributions can not be explained by cosmological redshift and time dilation
effects. Rather, broad distributions of total blastwave energies and bulk
Lorentz factors are necessary to explain the observed distributions
simultaneously within the framework of our unifying GRB model. We discuss
implications for source parameter distributions and determine a range of burst
parameters consistent with the data.
