We determine the integers a, b >= 1 and the prime powers q for which the word map w(x, y) = x(a)y(b) is surjective on the group PSL(2, q) ( and SL(2, q)). We moreover show that this map is almost equidistributed for the family of groups PSL(2, q) ( and SL(2, q)). Our proof is based on the investigation of the trace map of positive words.