$\lambda\propto L_\mathrm{bol}/{\mathcal{M}_*}$ denotes the *specific accretion rate*, the rate at which mass is accreted by the SMBH (traced by the AGN bolometric luminosity, $L_\mathrm{bol}$) scaled by the total stellar mass of the host galaxy, [${\mathcal{M}_*}$]{}, and $p(\lambda {\;|\;}{\mathcal{M}_*},z)$ is the probability density per logarithmic interval in $\lambda$ of a galaxy at a given redshift hosting an AGN with specific accretion rate, $\lambda$. We measured the slope of the probability distribution function as $\gamma=-0.65\pm0.04$, a strong redshift dependence ($\beta=3.5\pm0.5$) [[ and a normalization factor $\log_{10} A=-3.15\pm0.08$]{}]{}. Our work spanned a wide range of stellar masses ($9.5<\log {\mathcal{M}_*}/{\mathcal{M}_{\sun}}<12$) and focussed on low-to-moderate luminosity AGNs ($42<\log {L_\mathrm{X}}\;/\;\mathrm{erg\;s^{-1}}<44$). Thus our results spanned a limited range in specific accretion rate, $-3\lesssim\log \lambda\lesssim-1$, where $\lambda$ is defined relative to the Eddington limit assuming a single scaling between SMBH mass and the total stellar mass of the host galaxy [[[${\mathcal{M}_\mathrm{bh}}\approx 0.002{\mathcal{M}_*}$, based on @Marconi03 where we also assume ${\mathcal{M}_*}\approx \mathcal{M}_\mathrm{bulge}$]]{}]{}, $$\lambda=       \dfrac{L_\mathrm{bol}}{L_\mathrm{Edd}} 
= \frac{L_\mathrm{bol}}{ 1.3\; \times \; 10^{38} \; \mathrm{erg\; s}^{-1} \times 0.002 \dfrac{{\mathcal{M}_*}}{{\mathcal{M}_{\sun}}}}.
\label{eq:ledd}$$ These findings form the basis of our study in this paper. We investigate different models (see Section \[sec:model\] below) to extend the $\lambda$ distribution to higher accretion rates and compare with the 