  The shape of the probability distribution function (PDF) of molecular clouds
is an important ingredient for modern theories of star formation and
turbulence. Recently, several studies have pointed out observational
difficulties with constraining the low column density (i.e. Av <1) PDF using
dust tracers. In order to constrain the shape and properties of the low column
density probability distribution function, we investigate the PDF of multiphase
atomic gas in the Perseus molecular cloud using opacity-corrected GALFA-HI data
and compare the PDF shape and properties to the total gas PDF and the N(H2)
PDF. We find that the shape of the PDF in the atomic medium of Perseus is well
described by a lognormal distribution, and not by a power-law or bimodal
distribution. The peak of the atomic gas PDF in and around Perseus lies at the
HI-H2 transition column density for this cloud, past which the N(H2) PDF takes
on a powerlaw form. We find that the PDF of the atomic gas is narrow and at
column densities larger than the HI-H2 transition the HI rapidly depletes,
suggesting that the HI PDF may be used to find the HI-H2 transition column
density. We also calculate the sonic Mach number of the atomic gas by using HI
absorption line data, which yields a median value of Ms=4.0 for the CNM, while
the HI emission PDF, which traces both the WNM and CNM, has a width more
consistent with transonic turbulence.
