  Consider a convex body $C \subset \mathbb{R}^d$. Let $X$ be a random point
with uniform distribution in $[0,1]^d$. Consider the value $X_C$ equal to the
number of lattice points $\mathbb Z^d$ inside the body $C$ shifted by $X$. It
is well known that $\mathbb E X_C = \mathrm{vol}(C)$. The question arises: what
can be said about the variance of this random variable? This paper answers this
question in the case when $C$ is a polygon with vertices at integer points.
Moreover, an explicit distribution of $X_T$ is given for the integer triangle
$T$.
