Best probability textbook for self-study?

[galois]

I just spoke to a professor at a program of interest and asked what I should do to best prepare for my coursework but also the type of research that professor does. The most important thing emphasized was to know my probability very well. I never even ended up taking a formal probability class in undergrad! So I'd like to know if anyone has experience with a textbook that is particularly good for self-study.

I tend to like books where the exercises are proofs to write; I'm not sure if this will be the same in probability, perhaps it is more of a computing type of problem solving, but thought I'd mention it. For example, I'm currently reviewing linear algebra via "Linear Algebra Done Right" by Sheldon Axler and it is fantastic.

Also, if someone has a good source for calculus problems that would be great. I think instead of a textbook I'd rather just have a lot of tough derivative/integral problems and work through them until I remember all the relevant bits.

[Cavalerius]

I don't know whether it's the best, but I recently picked up Resnick's A Probability Path and find its presentation to be very lucid. Here is a link to its table of contents: https://d-nb.info/955671957/04.

If you are looking for a more mathematical book that deals with abstract measure theory, this book might not be what you need, however, since it is geared toward graduate students in statistics, applied probability, biology, operations research, mathematical finance and engineering, rather than in pure mathematics.

Edited March 20, 2019 by Cavalerius

[bayessays]

Resnick's book is pretty good, but it actually *is* a very mathematical, measure-theoretic probability book.  If you're preparing for the first year, this is probably overkill.  I'm guessing you would like something that will help prepare you to succeed in your first semester Casella & Berger class, especially if you have not had any probability course at all.  I'd take a look at Joe Blitzstein's Harvard Stat 110 class (https://projects.iq.harvard.edu/stat110).  There are lectures, homework assignments, tests, all available for free, and he is a genius at teaching the intuition behind everything, which you will really need.   It is an undergraduate course, but if you master the material, you will be set up for success in your Casella & Berger class.

As for calculus, I would just look up some integration by parts practice problems.  

[Cavalerius]

True, sorry, I did not see that you had never taken a probability class. I like Resnick's book, but it's probably too mathematical and not as helpful without some prior background in probability. At Duke, it seems that Resnick's book is used in the first semester for their probability requirement, and that is the reason that I picked it up. UT Austin doesn't formally require probability theory, but otherwise their program is similar to Duke's, so I am choosing to use my time before starting my PhD studies in the fall to learn some more probability on my own.  

Edited March 20, 2019 by Cavalerius

[Stat Assistant Professor]

1 hour ago, bayessays said:

Resnick's book is pretty good, but it actually *is* a very mathematical, measure-theoretic probability book.  If you're preparing for the first year, this is probably overkill.  I'm guessing you would like something that will help prepare you to succeed in your first semester Casella & Berger class, especially if you have not had any probability course at all.  I'd take a look at Joe Blitzstein's Harvard Stat 110 class (https://projects.iq.harvard.edu/stat110).  There are lectures, homework assignments, tests, all available for free, and he is a genius at teaching the intuition behind everything, which you will really need.   It is an undergraduate course, but if you master the material, you will be set up for success in your Casella & Berger class.

As for calculus, I would just look up some integration by parts practice problems.  

I second this. Your PhD program should teach you measure-theoretic probability, and there is not really any need to know it before starting (although I've found that a handful of PhD students, especially international ones, have already taken this class before). It may help to review undergraduate probability though. 

For Calculus, I would also recommend reviewing u-substitution and change of variables (for both single and double integrals) and derivatives for univariate functions (like chain rule, product rule, etc.) and partial derivatives. It's not at all necessary to review things like washer methods or any Calculus involving trigonometry or polar coordinates. You should be able to find "cheat sheets" for common differentiation and integration rules online, and it might help to do some practice using those (skip anything with trig).

[Bayequentist]

For Calculus: Schaum's Outline of Calculus is a classic book that has more than a thousand practice problems in it. The 5th edition is better than the 6th edition.

For Probability, Joe Blitzstein's class is awesome - I watched all the lectures and did most of the homework problems, so I can attest to the high quality of his course. If you still want a probability textbook, the standard recommendation would be A First Course in Probability by Ross. Though personally I've found Jaynes' Probability Theory: The Logic Of Science to be a much more enjoyable read. Jaynes is not strictly an introductory textbook, but if you've taken Real Analysis before then you should have no problem reading it, as he develops everything from the ground up.

[galois]

I have typically learned math through textbooks but that stat110 looks like a very convenient format, especially given my time constraints, so I'll probably do that. Thanks for the advice.

For Schaums, I imagine since it's so popular, even if answers aren't in the back of the book, they're probably easy to find online?

[Bayequentist]

4 minutes ago, galois said:

I have typically learned math through textbooks but that stat110 looks like a very convenient format, especially given my time constraints, so I'll probably do that. Thanks for the advice.

For Schaums, I imagine since it's so popular, even if answers aren't in the back of the book, they're probably easy to find online?

Schaums has 2 sections: Solved Problems and Supplementary Problems. In Solved Problems, the solution is shown right after the problem statement. In Supplementary Problems, they only provide the answers to the problems without showing you how to solve them. I'd imagine you'd be able to find a lot of solutions to the Supplementary Problems online, but don't quote me on that:P

[galois]

Hey all, first off just want to say thanks again for the Stat 110 suggestion - it has been fantastic! I'm on course to finish it before the semester starts, and I wanted to reach out for other suggestions. I have also purchased the Schaum Calculus book as suggested (and have been slowly motivating myself to do exercises).

I was trying to think of other things I should quickly prep before school, since it's been a while. I was thinking a review of computational linear algebra and real analysis. Anyone have recommendations for something I could squeeze into ~ 1 or 2 months part time? I just came across All the Mathematics You Missed on amazon, seems geared towards entering Math PhD, but curious if anyone has opinions on it.

Also, is Stat 110 a typical "probability" undergrad class, or "mathematical statistics"? These seem to be separate courses at most undergrad institutions, and I'm missing the context of where this lies in a typical math/stat education.

[bayessays]

Stat 110 is a typical undergraduate probability course - your first semester Casella Berger course in grad school is just a slightly harder version.  Mathematical statistics is the next chapters of the book - probability predicts how a model will generate data, mathematical statistics builds on those tools to go backwards, figuring out model parameters that fit given data.

Anything more than reviewing basics of linear algebra, how to do a convergence proof in real analysis, and integration by parts will be overkill.

Grad school is not about taking and passing classes - you are going to pass your classes unless you don't show up for the tests.  You will never know every piece of mathematics, so preparing for it by thinking you can just accumulate all the knowledge before it doesn't make sense.  I, like you, tried to learn all this stuff before grad school (and during grad school for future, and learning new programming languages I might need in future). It was all a waste of time.

That being said, if you like reviewing math, go for it! But I don't think you need to be provided specific books because you already, this moment, know what you need to succeed or they wouldn't let you in.  If you feel the need to do something "productive" to prepare, spending that hour a day learning some Chinese to help make friends with half your future classmates, or getting into some kind of exercise routine BEFORE school starts, will do way more for you than learning this one weird linear algebra trick.

[Bayequentist]

Aside from Mathematical Statistics, a course on Stochastic Processes is also a common follow-up to Stat 110.

Since galois seems to prefer watching lecture videos to reading books, fast.ai has a pretty interesting course on computational linear algebra.

[bayessays]

To bounce off bayequentist (who rightfully recommended the greatest probability book ever by Jaynes), if you want to learn something about stochastic processes, check out Resnick's Adventures in Stochastic Processes.  His other probability book is a standard textbook, but Adventures has an amazing sense of humor.

[galois]

I don't actually typically like watching lecture videos, but I'm enjoying this one. It may just be the nice separation of Strategic Practice / HW that I find so encouraging for self-study. Anyway @bayessays thanks for the advice, I think I needed to hear that. I've been stressing out, but in reality I should probably just chill out and enjoy free time while I have it. Maybe I'll just check out the fast.ai course casually once I'm done with 110. Thanks all!