  We derive the distribution of neutral and ionized gas in high redshift clouds
which are optically thick to hydrogen ionizing radiation, using published data
on Lyman-limit and Damped Lyman-alpha absorption systems in the redshift range
1.75 < z < 3.25. We assume that the distribution of the hydrogen total (HI+HII)
column density in the absorbers follows a power law K N_H^{-alpha}, whereas the
observed HI column density distribution deviates from a pure power law as a
result of ionization from a background radiation field. Comparison of the
models and observations give Maximum Likelihood solutions for the exponent
alpha and for X, the value of log(N_H/N_HI) when the Lyman-limit optical depth
is unity: alpha=2.7^{+1.0}_{-0.7} and X=2.75\pm0.35. X is much lower than what
would be obtained for a gaseous distribution in equilibrium under its own
gravity but the ratio of dark matter to gas density is not well constrained
being log(eta_0)=1.1\pm 0.8. An extrapolation of our derived power law
distribution towards systems of lower column density, the Lyman-alpha forest,
favours models with log(eta_0) < 1.1 and alpha=2.7-3.3. With alpha appreciably
larger than 2, Lyman-limit systems contain more gas than Damped Lyman-alpha
systems and Lyman-alpha forest clouds even more. Estimates of the cosmological
gas and dark matter density due to absorbers of different column density around
z=2.5 are also given.
