Pulsar motions in our Galaxy from their birth until 2 Gyr are studied statistically via Monte-Carlo simulation of 2 \times 10 ^ { 5 } pulsars with the best available representation of the Galactic potential .
We find that the distribution of height above the Galactic plane for pulsars with characteristic ages less than about 8 Myr could be well fitted by a Gaussian function .
For older pulsars , an extra exponential function is necessary to fit the distribution .
The scale-height of the Gaussian component increases linearly with time until about 40 Myr .
The height distribution becomes stabilized after about 200 Myr .
These results are not sensitive to initial height or radial distributions .
Taking the relationship between the initial velocity and height distribution , we found from the latest pulsar catalog that the height distribution of pulsars younger than 1 Myr directly implies the mean initial velocity of 280 \pm 96 km s ^ { -1 } .
Comparison of simulated sample of pulsars with the current available millisecond pulsars shows that their 1D initial velocity dispersion should be most probably 60 \pm 10 km s ^ { -1 } .