  An empirical model is presented that links, for the first time, the
demographics of AGN to their ensemble X-ray variability properties.
Observations on the incidence of AGN in galaxies are combined with (i) models
of the Power Spectrum Density (PSD) of the flux variations of AGN and (ii)
parameterisations of the black-hole mass vs stellar-mass scaling relation, to
predict the mean excess variance of active black-hole populations in
cosmological volumes. We show that the comparison of the model with
observational measurements of the ensemble excess variance as a function of
X-ray luminosity provides a handle on both the PSD models and the black-hole
mass vs stellar mass relation. We find strong evidence against a PSD model that
is described by a broken power-law and a constant overall normalisation.
Instead our analysis indicates that the amplitude of the PSD depends on the
physical properties of the accretion events, such as the Eddington ratio and/or
the black hole mass. We also find that current observational measurements of
the ensemble excess variance are consistent with the black-hole mass vs stellar
mass relation of local spheroids based on dynamically determined black-hole
masses. We also discuss future prospects of the proposed approach to jointly
constrain the PSD of AGN and the black-hole mass vs stellar mass relation as a
function of redshift.
