Advanced Probability Concepts

We finish our study of probability with a couple of slightly more technical ideas.

First, we look more carefully at discrete random variables and the role of countability. This helps explain exactly what distinguishes the discrete case from the continuous one, rather than relying only on familiar examples.

We then return to the idea of a random variable itself, giving a more formal account of what one actually is. This builds on the intuition developed earlier, while taking us a little closer to the rigorous definitions used in more advanced probability theory.

These ideas are not essential for every application in statistics or econometrics, but they help tidy up some important conceptual loose ends before the formal study of probability in measure theory.