  We present several results related to statistics for elliptic curves over a
finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages
of Euler products that we demonstrate. In this general framework, we can
reprove known results such as the average Lang-Trotter conjecture, the average
Koblitz conjecture, and the vertical Sato-Tate conjecture, even for very short
intervals, not accessible by previous methods. We also compute statistics for
new questions, such as the problem of amicable pairs and aliquot cycles, first
introduced by Silverman and Stange. Our technique is rather flexible and should
be easily applicable to a wide range of similar problems. The starting point of
our results is a theorem of Gekeler which gives a reinterpretation of Deuring's
theorem in terms of an Euler product involving random matrices, thus making a
direct connection between the (conjectural) horizontal distributions and the
vertical distributions. Our main technical result then shows that, under
certain conditions, a weighted average of Euler products is asymptotic to the
Euler product of the average factors.
