  We performed a numerical experiment designed for core formation in a
self-gravitating, magnetically supercritical, supersonically turbulent,
isothermal cloud. A density probability distribution function (PDF) averaged
over a converged turbulent state before turning self-gravity on is well-fitted
with a lognormal distribution. However, after turning self-gravity on, the
volume fractions of density PDFs at a high density tail, compared with the
lognormal distribution, increase as time goes on. In order to see the effect of
self-gravity on core formation rates, we compared the core formation rate per
free-fall time (CFR$_{\rm ff}$) from the theory based on the lognormal
distribution and the one from our numerical experiment. For our fiducial value
of a critical density, 100, normalised with an initial value, the latter
CFR$_{\rm ff}$ is about 30 times larger the former one. Therefore, self-gravity
plays an important role in significantly increasing CFR$_{\rm ff}$. This result
implies that core (star) formation rates or core (stellar) mass functions
predicted from theories based on the lognormal density PDF need some
modifications. Our result of the increased volume fraction of density PDFs
after turning self-gravity on is consistent with power-law like tails commonly
observed at higher ends of visual extinction PDFs of active star-forming
clouds.
