Lattice points inside a random shifted integer polygon

Consider a convex body C Rᵈ. Let X be a random point with uniform distribution in 0, 1ᵈ. Consider the value XC equal to the number of lattice points Zᵈ inside the body C shifted by X. It is well known that E XC = 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 XT is given for the integer triangle T.