Probability and Stochastic Systems

Description of Course Goals and Curriculum

1. ORF 309 is a course not as much in probability but rather in probability modeling. The purpose of the course is to teach you how to create models describing systems of uncertainty and how to quantify that uncertainty to help you make decisions. This course is a core course for ORFE majors and will establish fundamental skills required to take higher level courses in the department. 2. The course is structurally divided into two halves. The first half focuses on establishing foundations of probability and introducing the student to various kinds of random variables. This gives one a flavor for the simplest of probability processes and how to think about simple problems such as dice throws and coin flips. The second half focuses on modeling probability processes, which have more complicated structures. For example, you learn how to quantify uncertainty in random processes that evolve over time. 3. The course is chronological. Every lecture relies heavily on theoretical principles built in the previous classes. Since the models are built from ground-up, the course is also well suited for students with no background in probability. 4. The most challenging aspect of the course is that it forces you to think about the world in a different way. Most students taking this class are sophomores and are accustomed to the rather mechanical and computational nature of problem solving found in introductory quantitative courses like MAT 201, 202 and ORF 245. This course is built on a very different paradigm that requires you to have an open and creative approach towards problem solving. It lays equal emphasis, if not more, on higher level strategy involving setting up a model or framework as it does on the lower level computations required to solve the problem. 5. The “hidden” expectation in this course is the value of the precept. In many other courses, especially quantitative courses, students often overlook precepts presuming their ability to solve the math required through mere textbook guidance. This course requires you to attend precept and solve example problems to understand the aforementioned paradigm. Only then will you be able to solve homework and exam problems in a time-efficient manner.

Learning From Classroom Instruction

Lectures and Lecture Notes Prof. Van Handel teaches the lectures to ensure that you understand the required theory very well. His self-written notes complement the lectures and mirror the content almost exactly, which is why it is recommended that you read the lecture notes prior to attending the corresponding lecture. Don’t be fooled by the sample problems in the lectures and lecture notes. These are often much more simplistic than what is solved in precept, homework and exams. Precept The precepts have two important components: course “tricksters” and sample problems, both of which are equally essential to one’s success. The former refers to derived mathematical results that serve as tips and tricks to solve homework and exam problems. In lectures the professor is very focused on ensuring that you understand the theory, which is why he does not introduce too many corollaries. These corollaries, which I refer to as tricksters, are taught in precept and are essential to your time-efficiency and success in solving homework problems. The latter are an important bridge between the sample problems discussed in lecture and the more challenging homework problems. They use theorems from lecture and the tricksters previously introduced to solve problems of intermediate difficulty. These sample problems are often designed similarly to the homework problems for the week, which means that understanding how to solve them is key to solving the homework.

Learning For and From Assignments

Homework/Problem Sets The p-sets for this class are very time-intensive and it is advised that you start a few days before the deadline. Their purpose is to test your ability to apply probability theory and develop modeling skills to solve problems well within reach of real-world applications. The problems in each week are usually in order of difficulty, and the more difficult problems often require you to use the tricksters taught in precept. P-set groups and precepts are thus essential to solving these problems. Pedagogically, it is most effective to spend time thinking about and modeling a problem by yourself before working it out with your p-set group or going to office hours. Going through this process will ensure that you aren’t merely riding the wave of a better prepared p-set group or a helpful TA. Moreover, be sure to avoid the complacencies that arise when working with a p-set group, which include reliance on others to solve the problems, and not understanding problems despite writing up a solution because your group has figured it out. Exams Exam problems are like the homework problems in how they test your thinking though they may not be as computationally involved. There are 3 components to performing well in the exams: 1. Working through HWs and precept problems – These will help you know what kind of problems to expect on exams, the most frequently tested concepts and how to use tricksters in the context of a problem. 2. Making a great cheat sheet – The cheat sheet is critical to your success because the course covers a lot of content. Be sure to put the most important theorems and ALL tricksters learnt in precept on the cheat sheet. Also, put the toughest past exam questions you came across and be sure to organize these by topic for quick access. 3. Solve past exams diligently – Firstly, be aware that different versions of this class were taught by different professors. Older past exams were likely in years taught by Prof. Cinlar. These focused more on computation and less on probability modeling. Use these as practice to test your understanding of basic and intermediate concepts in guided problems. The more recent exams were in years taught by Prof. Van Handel. These are the ones you want to focus on. Solve at least one of these in real exam circumstances – time yourself and find a quiet spot so that you can think.

External Resources

Google is a helpful resource in understanding how to solve problems that are similar to homework problems. Be aware of the pedagogical effects though, and know that it is best advised to have a strong personal attempt before you resort to this.

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

1. This course enables you to take a lot of higher level ORFE classes, such as ORF 335 (Financial Mathematics) and ORF 405 (Regression and Applied Time Series), which apply probability to interesting fields. Moreover, you develop knowledge that helps you in various other classes, from Statistics to Computer Science to Mathematical Finance. 2. Be ready to invest a fair amount of time in this course. Problem set time commitment ranges from perhaps 4-5 hours early on to 10-15 later. The final problem set, which you have the entire winter break to complete, will take much longer. Be sure to start early and consult with office hours often enough. 3. Besides the obvious, which includes knowledge of probability and probability modeling, the most valuable lessons from this class are work habits. You learn how to work regularly with a group of equally skilled peers (your study group). You learn how to make an effective cheat sheet (this will be useful to ORF majors for future classes). You learn how to think about the world from a probabilistic lens, which is perhaps the most valuable lesson that will sub-consciously and consciously help you make many professional and personal decisions in the years to come. 4. Reiterating, be sure to focus more on modeling than computation. Setting up a problem is the more difficult and more important part of solving it, especially in today’s era of automated computation. Not only will this help you get a better grade but will also benefit you pedagogically.