terms of specific accretion rate, $\lambda$ (see Figure \[fig:accdists\], center). The tail in $p(\lambda {\;|\;}{\mathcal{M}_*},z)$ at high specific accretion rates ($\log \lambda>0$) can be attributed to the probability of finding galaxies with elevated SMBH masses relative to the host (due to the intrinsic scatter) accreting close to their Eddington limits.

Figure \[fig:xlf\_modB\] compares the observed XLF and our prediction using model B. The softer cutoff in the $\lambda$ distribution due to the scatter of the [${\mathcal{M}_\mathrm{bh}}$]{}–[${\mathcal{M}_*}$]{} relationship results in a smoother break in our predicted XLF compared to model A (Figure \[fig:xlf\_modA\]). Furthermore, at the bright end of the XLF (${L_\mathrm{X}}\gtrsim10^{44.5}$ [$\mathrm{erg\;s^{-1}}$]{}), AGNs with typical mass SMBHs are outnumbered by the small fraction of lower stellar mass galaxies with elevated SMBH masses, due to the intrinsic scatter in the [${\mathcal{M}_\mathrm{bh}}$]{}-[${\mathcal{M}_*}$]{} relation and the steep slope of the SMF at high stellar masses [${\mathcal{M}_*}\gtrsim10^{11}{\mathcal{M}_{\sun}}$, see also @Somerville09]. This effect leads to a somewhat flatter predicted bright-end slope than found with Model A. Nonetheless, our prediction still declines more rapidly than the observed XLF at $L_\mathrm{X}\gtrsim 10^{45}$[$\mathrm{erg\;s^{-1}}$]{}, especially at high redshifts ($z\gtrsim0.5$).

Model C
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Prompted by the lack of agreement at the bright end of the XLF using model B, our 