  We calculate the one-point probability distribution function (PDF) for cosmic
density in non-linear regime of the gravitational evolution. Under the local
approximation that the evolution of cosmic fluid fields can be characterized by
the Lagrangian local dynamics with finite degrees of freedom, the analytic
expressions of PDF are derived taking account of the smoothing effect. The
validity and the usefulness of the local approximation are then discussed
comparing those results with N-body simulations in a Gaussian initial
condition. Adopting the ellipsoidal collapse model (ECM) and the spherical
collapse model (SCM) as Lagrangian local dynamics, we found that the PDFs from
the local approximation excellently match the simulation results in the case of
the cold dark matter initial spectrum.
  As for the scale-free initial spectra given by $P(k)\propto k^n$,
  N-body result suffers from spurious numerical effects, which prevent us to
give a detailed comparison.
  Nevertheless, at the quality of N-body data, the model predictions based on
the ECM and the SCM quantitatively agree with N-body results in cases with
spectral index n<0.
  For the index n>=0, choice of the Lagrangian local dynamics becomes crucial
for an accurate prediction and a more delicate modeling is required, however,
we find that the model prediction based on the ECM provides a better
approximation to the N-body results of cumulants and PDFs.
