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We study the class AvgBPP that consists of distributional problems which can be solved in average polynomial time (in terms of Levin's average-case complexity) by randomized algorithms with bounded error. We prove that there exists a distributional problem that is complete for AvgBPP under polynomial-time samplable distributions. Since we use deterministic reductions, the existence of a deterministic algorithm with average polynomial running time for our problem would imply AvgP = AvgBPP. Note that, while it is easy to construct a promise problem that is complete for $\bf promise\mbox{-}BPP$ [Mil01], it is unknown whether BPP contains complete languages . We also prove a time hierarchy theorem for AvgBPP (there are no known time hierarchy theorems for BPP). We compare average-case classes with their classical (worst-case) counterparts and show that the inclusions are proper.