  The effect of kick velocity to newly born pulsars on the distribution of
$\gamma$-ray bursters is examined in the context of a disk origin model and a
halo model of $\gamma$-ray bursters. The conversion formula from a
2-dimensional velocity distribution function to a 3-dimensional distribution
function is derived and applied to reproduce the distribution function of the
kick velocity of radio pulsars. Monte Carlo simulations of the kicked neutron
stars show that the disk neutron star model of $\gamma$-ray bursters still
needs unnatural assumptions if the velocity distribution of the $\gamma$-ray
bursters is the same as that of neutron stars---only neutron stars with very
high kick velocities can become $\gamma$-ray bursters and there are silent
majorities. On the other hand, the core radius of $\gamma$-ray bursters is not
found to be extended by the kick velocity if the core-halo structure similar to
the Galactic dark matter distribution is based on the initial distribution of
the halo neutron stars. Thus the introduction of a kick velocity to neutron
stars do not improve the statistics of neither the disk model nor the halo
model. Two possibilities to save Galactic models are suggested: (1) The
$\gamma-$ray bursters are old neutron stars which were accelerated to
velocities faster than $750~{\rm km\; s^{-1}}$ by jet propulsion and passed the
death line for pulsars due to spin down from the rotational energy loss by jet
ejection. (2) The initial distribution of neutron stars is fairly uniform and
the extent of the halo is large enough for the halo model to be consistent with
the observations. It implies that the initial star formation burst of the
Galaxy occurred fairly uniformly in the extended halo region.
