  We place empirical constraints on the physical properties of $\gamma-$ray
burst events at cosmological distances. In particular we derive probability
distributions for the radiation energy $E_\gamma$, the minimum Lorentz factor
$\gamma_{\rm min}$, the maximum baryonic mass $M_{\rm max}$ and the upper bound
on the surrounding gas density $n_{\rm max}$ in the events, based on 169 bursts
from the first BATSE catalog. Using peak flux as a distance indicator we probe
bursts where the constraints are stronger than average. The resulting variance
and skewness of the cosmological probability distributions are calculated in
addition to their mean values: $\langle E_\gamma\rangle=4\times 10^{51} h^{-2}
{\rm erg}$, $\langle \gamma_{\rm min}\rangle= 5\times 10^2$, $\langle M_{\rm
max}\rangle=10^{-5}\xi^{-1} M_\odot$, and $\langle (\gamma/\gamma_{\rm
min})^{-5} n_{\rm max}\rangle\lsim 10^{2} \xi^{-1}{\rm cm^{-3}}$, where $\xi$
is the fraction of the total energy which is converted to $\gamma-$rays. The
distribution of burst energies ends at about $10^{53} {\rm erg}$, close to the
binding energy of a neutron star.
