  We extend our results in \cite{hislop_marx_1} on the quantitative continuity
properties, with respect to the single-site probability measure, of the density
of states measure and the integrated density of states for random Schr\"odinger
operators. For lattice models on $\mathbb{Z}^d$, with $d \geq 1$, we treat the
case of non-compactly supported probability measures with finite first moments.
For random Schr\"odinger operators on $\mathbb{R}^d$, with $d \geq 1$, we prove
results analogous to those in \cite{hislop_marx_1} for compactly supported
probability measures. The method of proof makes use of the Combes-Thomas
estimate and the Helffer-Sj\"ostrand formula.
