4.2 due on 10/17

Difficult
This reading introduced a few new concepts, leaving me with lots of questions. First off, I understand the definition of a probability mass function but I'm having trouble applying it. I'm running into the same problem as before: I don't know where the numbers come from. In the poker example, I know the numerator is all the outcomes which satisfy 2 pairs, but I don't know how to get that numerator. I also thought the equations for a proper random variable and the Key Rule looked similar. What's the difference between the two? And what's an example of where the sum of probability mass functions is less than one?
I also did not fully understand the Poisson Distribution and the Negative Binomial Distribution. For Poisson, I don't see how the sum equals 1. I referred back to Theorem 3.6.9, but that only confused me more. For the Negative Binomial, how do we decide what f(r) equals? And how do we know f(r) lies within [0,1]? The cumulative distribution function F(x) completely confused me. I don't understand what it does or its relationship to f(x). Figure 4.1 didn't really help either. Is F(x) like finding the integral of f(x)?
Reflective
Overall, I feel like the book is skipping steps and I'm having trouble filling in the missing pieces. I feel like I should be able to figure out some of these steps on my own but I can't. I had the most trouble in this reading with the cumulative distribution function. I want to say that F(x) is like finding an integral since it's a distribution, but I'm still not sure.