  We present here a simple model for the star formation history of galaxies
that is successful in describing both the star formation rate density over
cosmic time, as well as the distribution of specific star formation rates of
galaxies at the current epoch, and the evolution of this quantity in galaxy
populations to a redshift of z=1. We show first that the cosmic star formation
rate density is remarkably well described by a simple log-normal in time. We
next postulate that this functional form for the ensemble is also a reasonable
description for the star formation histories of individual galaxies. Using the
measured specific star formation rates for galaxies at z~0 from Paper III in
this series, we then construct a realisation of a universe populated by such
galaxies in which the parameters of the log-normal star formation history of
each galaxy are adjusted to match the specific star formation rates at z~0 as
well as fitting, in ensemble, the cosmic star formation rate density from z=0
to z=8. This model predicts, with striking fidelity, the distribution of
specific star formation rates in mass-limited galaxy samples to z=1; this match
is not achieved by other models with a different functional form for the star
formation histories of individual galaxies, but with the same number of degrees
of freedom, suggesting that the log-normal form is well matched to the likely
actual histories of individual galaxies. We also impose the specific star
formation rate versus mass distributions at higher redshifts from Paper III as
constraints on the model, and show that, as previously suggested, some galaxies
in the field, particularly low mass galaxies, are quite young at intermediate
redshifts. As emphasized in Paper III, starbursts are insufficient
...[abridged]
