  We study the effects of rounding on the moments of random variables.
Specifically, given a random variable $X$ and its rounded counterpart
$\operatorname{rd}(X)$, we study $|\mathbb{E}[X^k] -
\mathbb{E}[\operatorname{rd}(X)^{k}]|$ for non-negative integer $k$. We
consider the case that the rounding function $\operatorname{rd} :
\mathbb{R}\to\mathbb{F}$ corresponds either to (i) rounding to the nearest
point in some discrete set $\mathbb{F}$ or (ii) rounding randomly to either the
nearest larger or smaller point in this same set with probabilities
proportional to the distances to these points. In both cases, we show, under
reasonable assumptions on the density function of $X$, how to compute a
constant $C$ such that $|\mathbb{E}[X^k] -
\mathbb{E}[\operatorname{rd}(X)^{k}]| < C\epsilon^2$, provided
$|\operatorname{rd}(x) - x| \leq \epsilon \: E(x)$, where $E : \mathbb{R} \to
\mathbb{R}_{\geq 0}$ is some fixed positive piecewise linear function. Refined
bounds for the absolute moments $\mathbb{E}[ |X^k-\operatorname{rd}(X)^{k}|]$
are also given.
