  The jet opening angle theta_jet and the bulk Lorentz factor Gamma_0 are
crucial parameters for the computation of the energetics of Gamma Ray Bursts
(GRBs). From the ~30 GRBs with measured theta_jet or Gamma_0 it is known that:
(i) the real energetic E_gamma, obtained by correcting the isotropic equivalent
energy E_iso for the collimation factor ~theta_jet^2, is clustered around
10^50-10^51 erg and it is correlated with the peak energy E_p of the prompt
emission and (ii) the comoving frame E'_p and E'_gamma are clustered around
typical values. Current estimates of Gamma_0 and theta_jet are based on
incomplete data samples and their observed distributions could be subject to
biases. Through a population synthesis code we investigate whether different
assumed intrinsic distributions of Gamma_0 and theta_jet can reproduce a set of
observational constraints. Assuming that all bursts have the same E'_p and
E'_gamma in the comoving frame, we find that Gamma_0 and theta_jet cannot be
distributed as single power-laws. The best agreement between our simulation and
the available data is obtained assuming (a) log-normal distributions for
theta_jet and Gamma_0 and (b) an intrinsic relation between the peak values of
their distributions, i.e theta_jet^2.5*Gamma_0=const. On average, larger values
of Gamma_0 (i.e. the "faster" bursts) correspond to smaller values of theta_jet
(i.e. the "narrower"). We predict that ~6% of the bursts that point to us
should not show any jet break in their afterglow light curve since they have
sin(theta_jet)<1/Gamma_0. Finally, we estimate that the local rate of GRBs is
~0.3% of all local SNIb/c and ~4.3% of local hypernovae, i.e. SNIb/c with
broad-lines.
