What does the expected distribution of p-values look like? I suppose an answer should consider random hypothesis separately from those that are tested because there was some indication they might be true.
Some people model the distribution of p-values as a mixture of two distributions - called the BUM model.
The false positives are modelled as a uniform distribution. I think that this is the equivalent of the "random hypothesis" in your question.
The peak of p-values that occur for true positives is modelled as a beta distribution. The beta distribution can take lots of shapes and has to have its parameters fitted to match the collection of p-values.
My answer is from memory but if you are interested there is more information in this clear paper:
The paper concentrates on microarrays which provide interesting data when doing a lot of hypothesis testing - you do a lot of hypothesis testing but typically choose 1 p-value as a cutoff for all the testing.
