8.3 due on 12/5

Difficult
I still don't feel fully confident about change of variables, so I think it might be hard to find if RVs U = g(X) and V = h(Y) are independent. It's not so much the concept that gives me trouble but the lack of practice. In the Uniform Distribution example, I'm not sure about the reasoning behind the solution to part (a). The solution considers a set outside C and the intersection of a set with C. I thought we just had to look at sets of (x,y) in C. Also, in the Normal Densities example, I'm completely lost on part (c). I'm not even sure what the question is asking. Perhaps if there were a picture to show what we're trying to find the probability of?
Reflective
Even though we've gone over independence before, I feel like I might still have trouble with this section. The concept is the same, just the way you compute it is different. Before, we looked at events and discrete random variables. I believe the trick is to remember that for jointly distributed, we don't look at the density f (~ p.m.f.), but rather the distribution function F.