5.2, 5.11(a), (b) due on 11/5

Difficult
I don't see how Example 1 in section 5.2 relates to independent random variables. I'm assuming it's used to show how two marginal mass functions can be the same but the joint mass functions are different, but I'm not sure. In section 5.11 part (a), I don't follow solution II. I don't know what the event H includes. I thought it was the event a hole is halved, but isn't that what Z is measuring?
Reflective
The proof of Theorem 7(a) in section 5.2 reminds me of the double integral in calculus, only with double sums instead. I thought it was very similar; as long as the X variables are together and the Y variables are together, the equation works.