Summary form only given, as follows. The authors take the next step in developing the theory of average case complexity initiated by L. A. Levin. Previous work has focused on the existence of complete problems. The present authors widen the scope to other basic questions in computational complexity. Their results include: (1) the equivalence of search and decision problems in the context of average case complexity; (2) an initial analysis of the structure of distributional-NP under reductions which preserve average polynomial-time; (3) a proof that if all distributional-NP is in average polynomial-time then nondeterministic exponential-time equals deterministic exponential time (i.e., a collapse in the worst-case hierarchy); and (4) definitions and basic theorems regarding other complexity classes such as average log-space.