Intrinsic volumes, which generalize (and include) both Euler
characteristic and Lebesgue volume, are important properties of
$d$-dimensional sets.
We analyze and gives exact formulae for the expected value and
variance of the intrinsic volumes for a number of random models of
cubical complexes and show interleaving properties of their functions.
These values are useful for understanding shape of random
$d$-dimensional sets and for characterizing noise in applications.