Probability and Stochastic Systems

Description of Course Goals and Curriculum

ORF 309 is an undergraduate introduction course to probability and its applications, with a heavy emphasis on probability foundation. About half of the semester is dedicated to the basics (sample spaces, conditional probability, etc.), while the next half moves considerably faster through more advanced topics (Brownian motion, branching processes, Markov chains, etc.). The goal of this course is to be able to reason through probabilistic scenarios and determine what is the best way of approaching a probabilistic problem. By the end of this course, students should have a strong intuition for probabilistic scenarios, and have knowledge about different distributions and models, as well as their appropriate applications.

Learning From Classroom Instruction

There are three main teaching resources given in this class: classroom lecture, precept, and Piazza.   The course is paired with a set of lecture notes, which Professor Racz follows extremely closely. This means that anything you missed in class or are confused about will be explained very similarly in the lecture notes, even down to the examples for each subject, so it helps to review them before and after the class. The lecture notes are generally very clear and easy to follow, and Professor Racz is also very clear during his lectures. For each topic discussed in lecture, he will generally start with a scenario or a definition, and then follow with a theorem and its proof. The remainder of the discussion is then continued with example problems (that are in the lecture notes as well). Each lecture is paced normally, and Professor Racz will always stop to answer any questions and make sure students have a complete understanding of the topic being discussed. He is also available after the lecture to answer any lingering questions, which can be extremely helpful for any topics that students are still confused about or any supplemental questions students might have.   Precepts were run by graduate students, and they were either a structured lecture-type session on topics not covered in depth during class or in the lecture notes, or a freeform problem session with a short set of problems relating to the topics lectured on this week. The problems always have solutions posted a few days afterwards on Blackboard, and they are very thorough. Precepts are especially valuable because they occasionally cover topics that are not touched on in class or in the lecture notes, and the problems are very helpful preparation for exams. The solutions to precept problems are also good sources for reviewing material, and they are always written very clearly.   Like many other classes, ORF 309 has an associated Piazza, where students are free to ask clarifying questions about the class material, problems given during precept, or any assignments. Precept leaders as well as students are in charge of answering each other’s questions, and Professor Racz will also answer some questions. Answers are usually posted within one or two days, except before exams.   In general, the course moves very slowly for the first half of the semester but picks up pace by many factors for the second half. For this reason, getting very well acquainted with the material during the first half of the course is very crucial to doing well in the second half, because much of the material in the latter half will depend on concepts and intuition developed in the first half, and students are expected to pick these more advanced topics up very quickly.

Learning For and From Assignments

Problem sets in this class require using the material learned in class in a very intelligent or ingenuine way—often so that they seem very hard at first. For the first three or four weeks, the material being covered in class will be very approachable, so many people work individually. However, it helps a lot for students to collaborate during the second half of the semester on assignments, as problem sets get considerably harder—even harder than most of the problems on exams. Expect to spend at least 10 hours on some of these later problems sets. Most students will work together during office hours, which are a good time to ask preceptors questions. They are released on Thursday and due each Wednesday, and cover the material from the last week before the Wednesday they are due. Working on them early on may not be completely feasible as some of the relevant material may not be covered yet at the time.   The class consists of three assessments: two midterms, and a final exam. The two midterms are very speed-based; they are in class and have around ten problems each. This means that many people struggle to solve all of them. However, they are a good opportunity to see how well you are probability intuition is developing, as the key to doing well on the midterm is to know the general direction of the solution to each problem very quickly. The final is much more knowledge-based, since it is a three-hour long final with fewer problems. For all exams, a double-sided handwritten cheat sheet is allowed. It is helpful to have formulas and the means, variances, etc. of each type of variable on the cheat sheet, since there is a good amount of information to remember, and on a time-based midterm it is extremely useful to have a quick reference on hand just in case.   The best way to prepare for exams is to attend all the precepts, to do all the precept problems, and to go over all past problem sets and exams. Precept problems will usually teach you the tricks that are commonly used to solve problems that seem hard initially. Using problem set problems will usually overprepare you, so they are a very good resource—however, sometimes the graders will be slow to return them, so be sure to use other resources as well. Past exams are best done under real exam conditions, after you have studied for a while. The solutions are always posted and are generally very clear.   Since there are three exams, not doing as well as you hoped for on one can always be made up for in a later exam.  

External Resources

Professor Racz and all the preceptors always hold office hours several days during the week. Professor Racz’s office hours are more useful for questions about the course material, whereas the office hours that the preceptors hold are very helpful for completing problem sessions. Oftentimes there will be a lot of open discussion that happens at preceptor’s office hours between students and preceptor, so they are a good way to ensure understanding and solving the homework problems.   Piazza is also a helpful resource for those who operate on schedules that clash with office hours. The Piazza will always be active before exams, and Professor Racz will always mark “good questions” for other students to see.   PSET groups are also a vital resource in this class—most problems are very hard towards the end of the course, and it is often easy to make silly mistakes while solving them. Having someone else to check through your reasoning (without sharing solutions) is key to doing well without too much headache. Some problems on PSETS will also seem like a pain to calculate—you can always check your answers approximately on Desmos or Wolfram Alpha, and in some cases these tools will be necessary to calculate the final answer.

What Students Should Know About This Course For Purposes Of Course Selection

This is a core course for ORF concentrators, so ORF majors have to take this class. However, over half the class will also consist of COS, MAT, ECO, and other majors. MAT students might find this course especially helpful for getting a sense of the applications of mathematics, and COS and ECO students will also find this course’s material particularly relevant to their own concentration. The course material is crucial to many technical jobs, and the class is very interesting as well, so I would highly recommend that anyone give it a try.   One other factor in this course to consider before choosing it is that it has a very different workload between the first half of the semester and the second half of the semester. The first half of the semester is focused on basics that are covered in ORF 245 as well, so it seems to move very slowly, whereas the second half accelerates considerably and covers advanced topics at a much faster pace. This class is definitely a good indicator of whether or not ORF is the right fit for you but is also one of the tougher core classes in the ORF department.   Strategies and Tips

• Be familiar with MAT 201 and MAT 202 before taking this. You will be expected to take multiple dimensional integrals very easily, and have knowledge of topics such as series, Jacobians, and matrix operations.
• Find a good PSET group! As stated before, discussing the problems with others can fill in crucial gaps in your own understanding, and will help you work through the problems much faster.
• Check the Piazza frequently—if you are unclear on the wording of a PSET or can’t understand part of a solution, people are usually quick to respond, or someone may have already asked the question already.
• Read through the precept problem solutions—they often contain tips or tricks to simplify some problems, which will be extremely helpful on the exams.