  Fundamental aspects of fluid dynamics are related to construction of
statistical models for incompressible Navier-Stokes fluids. The latter can be
considered either \textit{deterministic} or \textit{stochastic,} respectively
for \textit{regular} or \textit{turbulent flows.} In this work we claim that a
possible statistical formulation of this type can be achieved by means of the
1-point (local) velocity-space probability density function (PDF, $f_{1}$) to
be determined in the framework of the so-called inverse kinetic theory (IKT).
There are several important consequences of the theory. These include, in
particular, the characterization of the initial PDF [for the statistical model
${f_{1},\Gamma} ]$ . This is found to be generally non-Gaussian PDF, even in
the case of flows which are regular at the initial time. Moreover, both for
regular and turbulent flows, its time evolution is provided by a Liouville
equation, while the corresponding Liouville operator is found to depend only on
a finite number of velocity moments of the same PDF. Hence, its time evolution
depends (functionally) solely on the same PDF. In addition, the statistical
model here developed determines uniquely both the initial condition and the
time evolution of $f_{1}.$ As a basic implication, the theory allows the
\textit{exact construction of the corresponding statistical equation for the
stochastic-averaged PDF}and the \textit{unique representation of the
multi-point PDF}'s in solely in terms of the 1-point PDF. As an example, the
case of the reduced 2-point PDF's, usually adopted for the statistical
description of NS turbulence, is considered.
