  We present predictions for the one-point probability distribution and
cumulants of the transmitted QSO flux in the high redshift Lyman-$\alpha$
Forest. We make use of the correlation between the Lyman-$\alpha$ optical depth
and the underlying matter density predicted by gravitational instability theory
and seen in numerical hydrodynamic simulations. We have modelled the growth of
matter fluctuations using the non-linear shear-free dynamics, an approximation
which reproduces well the results of perturbation theory for the cumulants in
the linear and weakly non-linear clustering regime. As high matter
overdensities tend to saturate in spectra, the statistics of the flux
distribution are dominated by weakly non-linear overdensities. As a result, our
analytic approach can produce accurate predictions, when tested against N-body
simulation results, even when the underlying matter field has rms fluctuations
larger than unity. Our treatment can be applied to either Gaussian or
non-Gaussian initial conditions. Here we concentrate on the former case, but
also include a study of a specific non-Gaussian model. We discuss how the
methods and predictions we present can be used as a tool to study the generic
clustering properties of the \lya forest at high-redshift.
  With such an approach, rather than concentrating on simulating specific
cosmological models, we may be in the position to directly test our assumptions
for the Gaussian nature of the initial conditions, and the gravitational
instability origin of structure itself. In a separate paper we present results
for two-point statistics.
