Exact evaluation of second moments associated with some families of curves over a finite field

Let Fq be the finite field with q elements. Given an N-tuple Q∈FqN, we associate with it an affine plane curve CQ over Fq. We consider the distribution of the quantity q−#Cq,Q where #Cq,Q denotes the number of Fq-points of the affine curve CQ, for families of curves parameterized by Q. Exact formulae for first and second moments are obtained in several cases when Q varies over a subset of FqN. Families of Fermat type curves, Hasse-Davenport curves and Artin-Schreier curves are also considered and results are obtained when Q varies along a straight line.