7.4 due on 11/26

Difficult
The Pareto Density example confused me a little bit. Conceptually, I understand the expectation is infinity and why, but when I tried to work it out on my own, I got stuck doing the calculations. Also, the justification for using Defintion 1 really confused me. I understand that the book is using a discrete approximation, but once again I get confused when trying to do the calculations. It's frustrating because I want to be able to see it work out mathematically instead of just accepting these statements as fact. At the end of the section, Jensen's inequality is mentioned. What does it mean for a function to be convex? Also, Chebyshov's inequality is simply stated, but there's no proof. I can get it to work out mathematically, but what does it mean conceptually?
Reflective
The section was relatively easy to understand since we've done expectations twice already. When justifying why we use Definition 1, I know the book tried to show us mathematically why it works. But instead, I just try to remember that we're dealing with continuous random variables, so we can't sum over every possible value of x since P(X=x) = 0. Surely the expectation for all random variables is not 0! So we use integrals since that will give us a "sum" of all possible values of x without equaling 0 all the time.