This is a linkpost for http://arxiv.org/abs/1601.03411
I just read Jim Babcock's post https://agentfoundations.org/item?id=374 and realized how similar math oracles are to 0' algorithms and strengths are to scores.
The mean running time is usually not used in average-case complexity theory because it is not well-behaved under changing the model of computation. Your example is in the class . For a good overview of average-case complexity theory see Bogdanov and Trevisan.
Thanks for that info! Uniform distributions are, however, generally used in analysis of algorithms, which is a different field from complexity theory.