  We investigate the black hole mass function (BHMF) and the Eddington ratio
distribution function (ERDF), focusing on the intermediate-mass black holes
(IMBHs) with masses down to $M_{\bullet}\sim10^4 M_\odot$. Based on the active
galactic nuclei (AGNs) with a detected broad H$\alpha$ emission line, we
construct a sample of 14,242 AGNs at redshift $z<0.35$, including 243 IMBHs
with $M_{\bullet}<10^6 M_\odot$. By jointly modeling the BHMF and ERDF via the
maximum posterior estimation, we find that the BHMF peaks at $\sim$$10^{6}
M_\odot$ and exhibits a relatively constant value of
$10^{-4}\,\mathrm{Mpc^{-3}\,dex^{-1}}$ at the low-mass end. By comparing the
derived BHMF of type 1 AGNs with the galaxy mass function based on the updated
black hole mass--host galaxy stellar mass relation, we derive the active
fraction. We also determine the active fraction for all AGNs using the upper
and lower limit of the type 1 fraction. The active fraction decreases from
15%--40% for massive galaxies ($M_\star>10^{10} M_\odot$) to lower than
$\sim$2% for dwarf galaxies with $M_\star\sim10^8 M_\odot$. These results
suggest that the black hole occupation fraction is expected to be $\sim$50% for
low-mass galaxies ($M_\star\sim10^{8.5}$--$10^9 M_\odot$) if the duty cycle is
similar IMBHs and supermassive black holes.
