course
Probability Theory

This class is super challenging. With that said, I think Karatzas is a good instructor and I think I learned a lot. I had taken real analysis I but not II. The pace is fast so you may feel like you don't fully understand some topics, but I also think that's just a product of the difficulty of the course. One nice thing about the course is that the beginning is much more typical proofy prob theory (e.g. DCT and Borel-Cantelli) but the later part switches to Markov chains which is a nice change of pace.

Never, ever, ever take prof. brown (YES, YOU. take this warning seriously). His definition of teaching is reading handwritten slides that are completely illegible half the time. To add to the immense confusion one feels from the horrible mess of chicken scratch that is his lecture notes and practice problems, his documents often have mistakes that he only amends days after lecture (there is an abundance of pdfs on coursework with amendments/corrections to notes). In line with this, trying to find the docs u need on courseworks is like trying to find a needle in a really really large haystack. His psets and practice problems are near impossible without a group of people with a shit ton of free time to dissect each question's meaning. The only saving grace from this hellhole was our TA, who made the exams more reasonable than Brown's practice problems. Nevertheless, the severe lack of preparation I had for even the 'easier' questions made this class fucking impossible. God I hate columbia's statistics department, and this class made me hate it even more. Note: A lot of these problems stem from the online setting and prof. brown is a really really sweet guy, all things considered. For me, supportive and nice professors typically compensate for difficult content and poor teaching, but this was waaaaaaay past my threshold for that.

Please dont take this class. he's a fine person, it's just that the stats department at this school is so absurdly bad it's ridiculous.

I took Probability Theory with David Rios. Unlike his intro stats class, I thought he was a damn good teacher for Probability Theory. His slides were clear and he explained the material thoroughly. His Office Hours were great. It's a hard class, but that's content not teaching. Overall, his teaching for the course was a little below-average for Columbia professors but that's normal. My personal suspicion is that he is totally plastered in the intro stats classes. I noticed they appear to be on his off days and he always came in a total wreck. In the intro stats course lectures, he swayed side to side a lot, really seemed out of it, said hilarious nonsense, and was unintelligible. Probability Theory was the opposite. Most lectures Rios was wearing a suit, was totally lucid and knew his stuff. I suspect he just doesn't care about the introductory classes.

The worst professor I have ever had not only in Columbia but also throughout my entire education history. His class is the total opposite of clear-explained, and he doesn't speak English. Most of the international students speak better English than he does. The most important, he is so indifferent and so cold. He truly doesn't care about his students; he ignored all of my emails, which I consider extremely rude. I had a really really important issue to negotiate with him, but he ignored all of my emails. I could say that he ruined my life for now. I don't know if things will work out a few years after, but it is impossible for me to say anything good now.

Didn't show up half the classes because there is really a no point (ta makes the exams). Surprisingly, I got an A. I have no idea how I did for the final because he didn't submit the scores.

Karatzas is a captivating lecturer and an excellent professor. He gave us challenging problems and we discussed them in class. The subject material we covered was clear, interesting, and practical. Though the only prerequisite for this course is Analysis 1, in order to truly understand the subject matter one should take both Analysis 1 and 2. For example we discussed Lebesgue theory in depth. Unfortunately I took this course while taking Analysis 1. This meant that I was way out of my depth. Yet Karatzas made the course accessible to me. His notes, the textbook, and the outside readings he provided were excellent sources that helped me catch up on the necessary background. He grounded every question of probability in a real world example, so that students could understand concepts even if they didn't understand the abstract definition. I spent much of he semester worrying that I would fail the class. Karatzas as a policy did not release the mean and standard deviation on exams, but I'm sure my scores brought the class average way down. If you care about getting homework grades returned, Karatzas is not the professor for you. That said, he gave me a B. My guess is that he gave everyone at least a B, and that grades were mostly made up.

I should've listened to the other reviewers. It was clear that the professor didn't care much about the class. It was also clear that the professor, TA, and grader never communicated. Lectures were very theory-oriented, and the TA office sessions were about proofs. But the homeworks were mostly word problems and so were the exams. The averages on most of the tests were between 50-60%, and I don't even know why because the problems weren't very hard. I got a bad grade. The TA was a condescending jackass. The professor was a sweet guy but didn't care much. It was hard to know what to study, and I didn't make this class a priority, so I feel the pain now. Wait to take this class with someone else, or skip if you can.

If you can, try and do the 3000 level version of this course instead of the 4000 level one. The 4000 level one is open to graduate students, and it will royally screw the curve for you. Be prepared to marry the book. Forget your girlfriend, forget your boyfriend--the book is your new significant other. I can't promise a happy voyage for you either, however. The book is decent at best. Just know if you're going to do any learning in this class, it sure won't be from the teacher. Sure, she's nice. Sure, she tries. But in the end, she's just a poor teacher. Have a question? Raise your hand? Oh, don't bother speaking-- she knows what your thinking (except she doesn't). She will finish your questions for you, assume everything is good, then move on. "Okay, fine, I'll be dandy with the book," you may think. Too bad she isn't organized either. She'll grab a little material from this chapter, or that chapter, until the exam shows up and she can't even tell you (true story) which sections the midterm is on! ("Uh I think we just go up to Chapter 3... Just follow the note I posted!") Just be prepared to work. Hard. I got A's in Calc I, II, III, Linear Algebra, and Discrete Mathematics and had to withdraw from this course. If you can get another teacher, please do so.

This class was the worst experience at Columbia. The professor is unclear with his explanations, do not cover enough example problems in class, and runs late on schedule. Therefore, the last weeks were very rushed. Throughout the entire semester, I had to learn by myself from the textbook, which was also very hard to understand. Although I do not consider myself as a statistics genius, I do consider myself as a quick learner. However, the way the professor explained things made the material much harder than it actually is. The course brought my grades down, and deprived my motivation to study statistics. Do everything you can to avoid this class.

First let me start off by saying that this class is largely dependent on who you TA is, if you get Andrew Davison, I wish you all the luck in the world. The material covered in this class can be pretty confusing mostly due to the awkward wording and multiple ways of interpreting various questions. On tests it really comes down to trying to figure out what the TA was trying to ask, and building your answer around that assumption. Professor Lo, has his quirks, and isn't exactly the best professor I've ever encountered. Most of his lectures are quite confusing and are often a waste of time. Every once in awhile there are a few gems in there so best to show up at least 50% of the time. Without a doubt he cares a lot about the subject, but just can't seem to get a point across unless on the rare occasion he decides to talk about a real world example from research he has done. What he lacks in teaching skills though, he makes up for in understanding, as long as you are willing to honestly let him know how the class and tests are going for you. Otherwise, because of his incredibly hands off approach to this class you are going to get crushed. To illustrate this, I emailed him on a number of occasions and never heard back. Everything, including the tests, are left up to his TA. Why is that so important? Because when you get a super probability genius TA PhD candidate from Cambridge, you are in for a world of hurt on the tests. For both our midterms the class average hovered around a median of 50%. About 2/3 of the class scored a little below that mark, but there were some outliers. Many people had around a 30% - 40% on both midterms. Not a single person finished either midterm, and many only finished about a half of it or a little more in the 75 minute time span. Because professor Lo is so understanding though, he took our pain to heart and created a drastic curve, and the final ended up being slightly easier, but still low (or should I say Lo...) grades overall. Makes sure you know who your TA is for this class before committing. Ultimately, you should be ok though, if not a bit demoralized. Wikipedia and Chegg Study will be your friend, because the textbook isn't the best in helping fill the gaps in knowledge from the class.

I am sorry to say so but Prof. Lo is not really approachable. If you sent him emails, it would be a miracle that he replied. He never designed or graded the mid-terms himself, separating himself with his students, only to be told the mid-terms were not even related to what he expected us to know. I really see no point of spending more than 200 dollars per class listening to him reading the textbook, let alone the fact that he could not even read his own notes clearly. Maybe the only good thing about Prof. Lo's class is that he did do something after the exam to make you feel better, yet I am not sure whether he has well thought through his ideas before presenting it to the class. In sum, it may be a good idea to save some money and time, and learn this course by yourself. However, the Department of Statistics has made this one a compulsory course for anyone whose major has something to do with statistics; and Prof. Lo happened to be the only one who is teaching this course. So be prepared. Don't have too much expectation from this course. Or maybe, it is going to overturn your perception of what a Columbia education is like. If given a chance, I wish I had never taken this course. Best luck!

This was one of the most worthwhile, rewarding, and challenging classes I’ve taken at Columbia. We covered a ton of material and the pace was very intense. The course basically covers a lot of standard probability topics from a rigorous mathematical perspective. A good part of the course involved introducing measure theory and Lebesgue integration, of which some was familiar from the end of analysis 2. Marcel is obviously brilliant and understands the material very well. He told us on the first day that while there are a lot of great connections between the intuition and mathematics of probability, we were not expected to realize this overnight and it would probably take some time, but that things would eventually fall into place if we continued to struggle with the material. He was definitely right about this. Lectures were generally well-structured and organized, but very fast. Towards the middle of the class, there were a number of very long proofs, some of which took the entire class period. We didn’t follow a textbook for this class, and I think it would have been helpful if some lecture notes were posted online, because it is a lot of dense, notational-heavy material copy from the board. Homework assignments were interesting and well written, but also quite challenging. Most of them took me a long time to complete, but I definitely learned a ton from working through them. Having taken the analysis sequence before this course was definitely helpful, but I felt this class was a step up in difficulty from analysis.

Marcel Nutz is great. He has some pretty funny anecdotes about his French education and the apparent cultural differences between France and US. Also, his English is clear and his notation on the chalkboard is consistent and "almost surely" (lol) readable. He doesn't demand too much of students in lecture: For more difficult proofs he usually spells things out, and is willing to provide additional details when students ask for them. The class is great. He conceded at the beginning of the course that the notion of measure and random variables as functions may not be immediately intuitive, but that it would make sense as the course proceeded. He was right. After taking Analysis; in particular, Analysis II, this course may seem like an introductory course because much of the time is spent on introducing definitions and elementary results from measure theory. In this sense, the course is pretty straightforward. However, for those not very comfortable with pure math (cough engineers/physics students ;-p), this course may take a bit more work. As far as preparation goes, having a background in mathematical statistics or seeing an introduction to measure theory could be helpful. Great course if interested in stochastic processes.

Good class, good professor. Doesn't have the best blackboard style and isn't the best at giving broad or deep mathematical views but explains concepts clearly and thoroughly. I wouldn't praise Neel as enthusiastically as some other reviewers. I've taken several 4000 level math courses and as a teacher Neel is probably about average, a good instructor but nothing out of the ordinary. He's also a nice, approachable guy with a dry sense of humor. If you're at all interested in analysis or probability this is a worthwhile class to take. There is some overlap with Stat 4105, with the difference that the course emphasizes the theoretical apparatus behind probability instead of modeling and problem solving. You'll probably have a slightly easier time if you're comfortable with basic probability concepts, but that's not required and you could take the class without even knowing what a probability distribution is. It is important though that you've already finished the first semester of Real Analysis since the textbook assumes familiarity with basic topological notions, sequences/series, continuity etc. I guess you could get by without that knowledge but you would be confused by a lot of the material and probably wouldn't do well. Lastly, if you're thinking about trying the graduate analysis/probability sequence this class covers some of the same material at a much lower level, so it would make decent preparation.

Neel is an excellent professor who loves explaining the enigmatic features of probability theory to anyone willing to listen. His one downside is that he talks very fast during lecture, so it is the class's responsibility to ask questions to clarify the material on the board. Otherwise, I could not imagine a better instructor for this heavily theoretical, upper-level applied mathematics course.