  The winding number is a concept in complex analysis which has, in the
presence of chiral symmetry, a physics interpretation as the topological index
belonging to gapped phases of fermions. We study statistical properties of this
topological quantity. To this end, we set up a random matrix model for a chiral
unitary system with a parametric dependence. We analytically calculate the
discrete probability distribution of the winding numbers, as well as the
parametric correlations functions of the winding number density. Moreover, we
address aspects of universality for the two-point function of the winding
number density by identifying a proper unfolding procedure. We conjecture the
unfolded two-point function to be universal.
