Course Summary

In elementary probability theory, the fundamental object of interest is a collection of outcomes from a given experiment or observation (the so-called sample space $\Omega$). One then studies probability functions on this space: functions $\mathbb{P}:\wp(\Omega) \to [0,1]$ which associated a number between $0$ and $1$ to subsets of $\Omega$ according to some mild axioms, namely that $\mathbb{P}(\Omega)=1$ and that $\mathbb{P}$ is ``additive" on (countable) disjoint sets. Often this analysis is broken into two diametrically opposed cases, those in which the objects in question are discrete (in which a healthy dose of combinatorics is often what is needed to resolve questions), and those where they are instead continuous (in which case one uses the tools of calculus --- particularly integrals --- are the key integredient).

The first pass at probability, however, often passes over subtleties which become more glaring as one's mathematical sophistication grows: is it feasible to attach a probability to all subsets of $\Omega$ in a sensible way? how do probability functions behave under limits? is the bifurcation between continuous and discrete accurate? and how does one resolve probability functions which are modeled by non-integrable functions? In this course we give a proper treatment of probability by starting with a general approach --- measure theory --- that unifies the discrete and continuous cases. We then review the key facets of probability theory (random variables, expectation, etc.) in this light, and use it to answer questions which the naive theory cannot easily resolve.

Course Instructor

The professor for this class is Andy Schultz. His office is on the main floor of Clapp Library, room 255. Office hours will be held

You are highly encouraged to attend office hours, and you never need an appointment to do so. If these office hours don't fit with your schedule, contact the instructor so that he can either adjust when ``official" office hours are held or set up an appointment to help you outside of office hours. Please come to the professor's office or send him an email if you ever want to discuss material from the class or ask about homework problems!

You can contact the instructor at . Though he is always happy to receive emails from you with questions or concerns about the course, he can't guarantee that he'll be able to promptly reply to emails late at night or over the weekend. If you do contact the professor by email, please be sure to follow standard email etiquette. In particular, please make sure you include a greeting and signature and avoid abbreviations. If you're contacting him to ask about a problem, please be sure to specify what the problem asks (as opposed to asking something like ``I can't get problem 2 and need your help").