Probability and Stochastic Systems

Description of Course Goals and Curriculum

This course teaches you how to solve a diverse range of probability applications by giving you a solid foundation in the basics of probability theory (i.e. discrete/continuous random variables, expectation, conditioning etc.) and building upon that. By the end of this course, you should be really comfortable with different kinds of distributions (Exponential, Uniform, Binomial, Bernoulli, Geometric, Normal, Poisson, Gamma, Negative Binomial), as well as with more advanced techniques involving Poisson Processes, Random Walks, Brownian Motion, Markov Chains, and Branching Processes.  Ultimately, this course is about applications of theory, so expect to do a lot of problem solving rather than poring over proofs.

Learning From Classroom Instruction

You will learn from three lectures a week, where the Professor will teach you the theory and solve some basic problems involving the theory. Professor Shkolnikov’s lectures are generally quite clear and operate at a normal pace. Each lecture is directly based on comprehensive lecture notes that are provided to you. I suggest that you attend lecture, while taking notes, to get an initial understanding of the purpose and methodology behind the material at hand, and then supplement the holes in your knowledge with the lecture notes. These holes will invariably become apparent to you while working on a challenging PSET problem, so refer often to these lecture notes to build a strong foundation. In precept, which you will have weekly, you will focus on solving slightly more challenging problems (though not as difficult as the PSET problems). These precept problems are essential to working through assignments/assessments so it is important that you also understand how to solve these in order to succeed in the course. Generally, precept problems also appear in the lecture notes. The point being, look at – and more importantly, try to understand – the lecture notes!

Learning For and From Assignments

The hardest problems you will see appear on your weekly PSETs. I recommend that you set up a study group to work on the PSETs together. An ideal group would be composed of people at/around your level, so that you are not forced to do the majority of the work yourself, nor are you failing to learn the requisite material by relying on others. Expect to spend 5-15 hours a week (depending on the difficulty of the problem set) finishing the assignments. The course has plenty of office hours run by TAs where you can ask for hints to help you out with the problems, or discuss with other students. These are especially helpful to review conceptually difficult material – after all, with a better theoretical foundation, you will inevitably perform better on your assignments. Make sure that your answers are clear and that you show your work so that partial credit can be given. Also, it is perfectly OK if you still have trouble understanding how to solve a particular problem. Just ensure that you review the solutions when they are posted and consult the Professor or a TA so that you do eventually understand because if you can do the PSET problems, you can definitely do the assessment problems. These assessments consist of two midterms and a final. You are allowed a cheat sheet for both assessments, so make sure you make good use of this benefit. I recommend that you review the lecture notes covered by this particular exam, and write down any key formulas on your sheet. Of particular importance are the variances/expectations/densities of various random variables. Some formulas that might not be covered in the lecture notes relate to summation formulas (e.g. geometric sum, sum of (n choose 2), sum of first n^2 integers etc.) or some tricks you may have learned from the problem sets/precept problems that might be useful. These formulas can help trigger your memory, so having as much down as possible (without over-clustering the sheet) works to your benefit. Once you have your cheat sheet, practice by doing problems from past finals and review your past PSETs, making sure that you fill in your cheat sheet with any new material that might be helpful. This course can only test you on so much, so by doing more practice problems, you will have a better grasp of the key concepts/tricks that will almost certainly show up on the assessments. Also note that your midterm score is supplemented by your final score if you perform better on the latter, so don’t worry if you don’t meet your expectations initially.

External Resources

With lectures, lecture notes, precepts and office hours at your disposal, external resources are only marginally useful if you can find them. As I mentioned earlier, a study group for PSETs is pretty much a must, so that is the only external resource I whole-heartedly recommend. And don’t be scared to email/talk to your preceptors if you have a question.

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

As this is a core course for ORFE, you don’t really have much of a choice about taking this course as an ORFE major. Nevertheless, there’s a reason ORF 309 is a core course, as the topics covered therein are invaluable to future courses. For this reason, I recommend that you do your best to truly understand the material as it is a prerequisite for virtually all future ORFE courses. Even if you are not an ORFE major, I recommend that you consider taking this course if you are at all interested in being able to solve problems related to probability. This applies in particular to COS majors who want to delve into neural networks or math track ECO majors, but the course is interesting for any major. Since the first half of the semester overlaps for ORF 245 and ORF 309, I recommend that you take these courses at the same time. This would save you time and ensure that you truly grasp the material across both classes. The second half of the semester will have ORF 245 branch out into statistics, but this will be quite different. I have heard (from advisers and other students) that your experience in this course serves as a fair indicator of whether or not you should continue with ORFE, so if you are still on the fence, be sure to take this course sooner rather than later!