(Abridged) We complete the census of nuclear X-ray activity in 100 early type Virgo galaxies observed by the Chandra X-ray Telescope as part of the AMUSE-Virgo survey, down to a (3sigma) limiting luminosity of 3.7E+38 erg/s over 0.5-7 keV. The stellar mass distribution of the targeted sample, which is mostly composed of formally `inactive galaxies, peaks below 1E+10 M_Sun, a regime where the very existence of nuclear super-massive black holes (SMBHs) is debated. Out of 100 objects, 32 show a nuclear X-ray source, including 6 hybrid nuclei which also host a massive nuclear cluster as visible from archival HST images. After carefully accounting for contamination from nuclear low-mass X-ray binaries based on the shape and normalization of their X-ray luminosity function, we conclude that between 24-34% of the galaxies in our sample host a X-ray active SMBH (at the 95% C.L.). This sets a firm lower limit to the black hole occupation fraction in nearby bulges within a cluster environment. At face value, the active fraction -down to our luminosity limit- is found to increase with host stellar mass. However, taking into account selection effects, we find that the average Eddington-scaled X-ray luminosity scales with black hole mass as M_BH^(-0.62^{+0.13}_{-0.12}), with an intrinsic scatter of 0.46^({+0.08}_{-0.06}) dex. This finding can be interpreted as observational evidence for `down-sizing of black hole accretion in local early types, that is, low mass black holes shine relatively closer to their Eddington limit than higher mass objects. As a consequence, the fraction of active galaxies, defined as those above a fixed X-ray Eddington ratio, decreases with increasing black hole mass.
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