  Turbulence is essential for understanding the structure and dynamics of
molecular clouds and star-forming regions. There is a need for adequate tools
to describe and characterize the properties of turbulent flows. One-point
probability distribution functions (pdf's) of dynamical variables have been
suggested as appropriate statistical measures and applied to several observed
molecular clouds. However, the interpretation of these data requires comparison
with numerical simulations. To address this issue, SPH simulations of driven
and decaying, supersonic, turbulent flows with and without self-gravity are
presented. In addition, random Gaussian velocity fields are analyzed to
estimate the influence of variance effects. To characterize the flow
properties, the pdf's of the density, of the line-of-sight velocity centroids,
and of the line centroid increments are studied. This is supplemented by a
discussion of the dispersion and the kurtosis of the increment pdf's, as well
as the spatial distribution of velocity increments for small spatial lags. From
the comparison between different models of interstellar turbulence, it follows
that the inclusion of self-gravity leads to better agreement with the observed
pdf's in molecular clouds. The increment pdf's for small spatial lags become
exponential for all considered velocities. However, all the processes
considered here lead to non-Gaussian signatures, differences are only gradual,
and the analyzed pdf's are in addition projection dependent. It appears
therefore very difficult to distinguish between different physical processes on
the basis of pdf's only, which limits their applicability for adequately
characterizing interstellar turbulence.
