  This paper estimates the specific accretion-rate distribution of AGN using a
sample of 4821 X-ray sources from both deep and shallow surveys. The specific
accretion-rate distribution is defined as the probability of a galaxy with a
given stellar mass and redshift hosting an active nucleus with a certain
specific accretion rate. We find that the probability of a galaxy hosting an
AGN increases with decreasing specific accretion rate. There is evidence that
this trend reverses at low specific accretion rates, $\lambda<10^{-4}-10^{-3}$
(in Eddington units). There is also a break close to the Eddington limit, above
which the probability of an accretion event decreases steeply. The specific
accretion-rate distribution evolves such that the fraction of AGN among
galaxies drops toward lower redshifts. This decrease in the AGN duty cycle is
responsible for the strong evolution of the accretion density of the Universe
from redshift $z\approx1-1.5$ to the present day. Our analysis also suggests
that this evolution is accompanied by a decoupling of accretion events onto
black holes from the formation of stars in galaxies. There is also evidence
that at earlier times the relative probability of high vs low specific
accretion-rate events among galaxies increases. We argue that this differential
redshift evolution of the AGN duty cycle with respect to $\lambda$ produces the
AGN downsizing trend, whereby luminous sources peak at earlier epochs compared
to less luminous ones. Finally, we also find a stellar-mass dependence of the
specific accretion-rate distribution, with more massive galaxies avoiding high
specific accretion-rate events.
