When we calculate the average of anything: we add up (or integrate) the sum{ [all the things were taking the average of] * [the probability of getting that thing ]}.
The thing about the average I'm that curious about is for example: the average height of 5 people turns out to be say 6...
I think its because the potential is periodic then the Hamiltonian is too: H(x)=H(x+a), you can then sub this in directly and the translation operator now just acts on psi