It seems like it should be where is the given random variable.

Since are independent of , conditioning by just gives (the conditioning doesn't affect the 's by independence; it just fixes the value of ).Find by conditioning on N

At the first glance I didn't quite understand the information given in this problem. But after a while I wrote down what I believe is asked to solve in the problem

But this is as far as I can go, I know P(k) = p(1-p)^(k-1), and I'm clueless on what's

I want to say that is independ of , but I'm not sure whether it's true.