Expected Value of the number of elements in an intersection of two random subsets of a set

Dear all,

This is my first post to this group. My question is as follows:
Consider a universal set Z. let N=|Z| be the number of elements in this
set. Let A and B be two subsets which are selected uniformly at random
from all subsets of Z (excluding the empty set). Let |A intersection B|
be the number of elements in the intersection of these two random
subsets. How can we calculate the following expected value:

E{|A intersection B|}

Further more, how can we calculate this expected value when A and B are
sampled according to any given distribution e.g. normal distribution
(not necessarily uniform).

regards,
hassan

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