Say I have two infinite collections of $0/1$ random strings of length $n$ , where each digit is an independent Bernoulli RV, with parameter $p_i$ in the first collection and $q_i$ in the second, $i$:1...$n$.
Define $X_i$ to be a Bernoulli RV representing the absolute difference between the digits at location $i$ of a random sample of two strings, each string from a different collection.
Define $Y_i$ to be a Bernoulli RV representing the absolute difference between the digits at location $i$ of a random sample of two strings from the combined collection (with prior probability 0.5 for sampling each string from collection 1 and 0.5 from collection 2).
Define $Z_i$ to be a Bernoulli RV representing the absolute difference between the digits at location $i$ of a random sample of two strings from the same collection (with prior probability 0.5 for sampling a pair from collection 1 and 0.5 from collection 2).
Are the $X_i$ independent? [I think Yes]
Are the $Y_i$ independent? [I think No]
Are the $Z_i$ independent? [I think No]
If there is dependency in each case, what is the correlation?