Recommendation for calculus-based probability book for self-study ?

[2016biostat]

Hello, I just finished taking Linear Algebra this summer (after finishing my undergrad. degree), and now have a fairly large amount of time before possibly starting in a graduate program in biostats (if i'm so fortunate). 

 

With that in mind, would anyone be able to recommend a calculus-based probability book for self-study? From what i've read, this topic seems like it would be a worthwhile to investigate -- even if i'm not able to take a formal class in it prior to grad school. I'm not really sure what variety is available re: rigor, but I would prefer something relatively palatable (if that's an option), as my math background only covers calc I-III, linear algebra, and introductory stats.

 

Thanks

[TakeruK]

My undergrad degree required a Calculus based probability course and we used "Introduction to Probability Models" by Sheldon M. Ross. We used the 9th edition, but it seems like it's on the 11th edition now on Amazon: http://www.amazon.com/Introduction-Probability-Models-Eleventh-Sheldon/dp/0124079482/ref=sr_1_3?ie=UTF8&qid=1435707897&sr=8-3&keywords=sheldon+M+ross&pebp=1435707905227&perid=09M0FVB3C4SB6PWHKD9M

 

I really liked this book. It's very comprehensive (but our course didn't cover all of the book). I think this might be the kind of book you are looking for. This was a third year course for me, and at that time, I also had approximately the same math background as you (I also had ODEs and PDEs but they are not necessary here).

 

In grad school, I took (and now TA) Bayesian statistics so if you are into that as well, I highly recommend "Data Analysis: A Bayesian Tutorial" by Sivia (http://www.amazon.com/Data-Analysis-Bayesian-Devinderjit-Sivia/dp/0198568320/ref=sr_1_1?ie=UTF8&qid=1435708020&sr=8-1&keywords=sivia&pebp=1435708022354&perid=0XQF4FZ88RFQVQJMA298). This book is very short and concise so at some points, a lot of steps are skipped. Notation and concepts from linear algebra is used A LOT here to simplify the notation so it's good that it's recent for you (when I took this, my last Linear Algebra course was something like 4 years ago).

 

Edited to add: Sivia's book jumps right into Bayesian stats assuming you already have introductory statistics, so I think it's better to start with Ross and move onto Sivia. Also, Sivia has some C code / pseudo code that is pretty helpful to illustrate some of the more complicated material. I find that some concepts, such as Markov Chains, are so much more understandable when you see it in action rather than as a series of equations and proofs.

[svent]

Two common books are the one by Ross (First Course in Probability, not the book mentioned above) and the one by Pitman. I don't really recommend the one by Pitman for self-study. It's alright, but some things about it are strange so I'd only get it if you're taking a class with it.

I've read parts of Probability and Statistics for Computer Scientists by Baron. You may not be interested in CS, but it seems to be a great brief no-nonsense book from the few chapters I've read. I don't think there's too much CS for a broader audience either. There seem to be some examples with some 2^n's thrown in and what not, it's not full of silly gambling/card problems like most texts on the subject. (I always hated that aspect of combinatorics/probability classes -- I don't play cards or poker so lots of problems were foreign to me, regardless of the math involved.)

[TakeruK]

I haven't heard of the other book by Ross before so I searched for an online table of contents. It seems like the two books are quite similar.

 

First Course in Probability: http://www.pearsonhighered.com/bookseller/product/First-Course-in-Probability-A-9E/9780321794772.page

Introduction to Probability Models: http://store.elsevier.com/Introduction-to-Probability-Models/Sheldon-Ross/isbn-9780124081215/

 

However, it looks like First Course in Probability is more focused on the mathematics and theory while Introduction to Probability Models is more about applied problems. 

 

So, I think it might depend on what you want to get out of the self-study. I'd recommend Introduction to Probability Models if you are not that interested in things like "how does the exponential distribution come about?" and are looking for "okay, taking the exponential distribution as a given, what can you do with this?" I personally like this book because, as you can see from the Table of Contents, there is a short intro to basic concepts (Ch. 1-3) followed by how to apply various probability theories to problems in the physical sciences. 

 

I am going to guess that "First Course in Probability" is more about the math from the Table of Contents and reading how the publishers market this book vs. the other book (but again, not having seen the whole thing, maybe svent can provide more information). 

[svent]

Well I passed SOA Exam P essentially by using the Ross book I mentioned and not worrying about the details of coming up with the expectation/variance for the different distributions, etc., just taking all that stuff as a given. I passed the exam, but afterward, all the different distributions, when to use each one, different formulas for E[X], Var(X) got all jumbled in my head until I later took a probability class and studied it for real. For me at least I'm much more likely to remember those things if I actually do the calculations or proofs myself.

 

Edit: The applied book has stuff on brownian motion and markov chains. If you care about that stuff (and don't mind not learning some of the math details), then get that book. If you just want very strong in-depth knowledge of probability, then go for the book I mentioned. Or just get the probability for CS book I mentioned, which is very quick to read.

Edited July 1, 2015 by svent

[TakeruK]

Edit: The applied book has stuff on brownian motion and markov chains. If you care about that stuff (and don't mind not learning some of the math details), then get that book. If you just want very strong in-depth knowledge of probability, then go for the book I mentioned. Or just get the probability for CS book I mentioned, which is very quick to read.

 

The Markov Chain and Brownian motion stuff is why I recommended the "applied" Ross book. It has entire chapters dedicated to these topics while in the other book, these topics are sections within a single chapter. The applied book and the course I took did have some useful "math stuff" like learning about the moment generating functions and how to get the mean, variance, etc. from differentiating said functions (so, fewer things to memorize). It also teaches some useful things like if you have two random variables X and Y, how do you determine the distribution of another random variable, Z, that is a function of X and Y.

 

I guess the "applied" book is like a "toolbox" of probabilistic models and other tricks you can apply to whatever problem you are trying to solve. I don't consider myself an expert in probability theory at all, but instead, I feel that given a data-driven problem and a question, I know what tools to use to answer that question, and where to find those tools in the book. I certainly do not know how to derive/prove these theories myself, and I am not qualified to develop my own statistical methods; but I do know enough "math stuff" to follow along someone else's proof.

[sabq]

I know you asked for a book, but, I strongly recommend you check Coursera's mathematical biostatistics boot camp. You can also check out edx's offers. The MOOCS I completed so far are better than almost every course I took in a 'real' university.

[2016biostat]

Thanks for the speedy responses! I ordered the Ross: "Introduction to Probability Models" book to start off. That Coursera program looks interesting, too. 

Edited July 1, 2015 by 2016biostat

[svent]

The Markov Chain and Brownian motion stuff is why I recommended the "applied" Ross book. It has entire chapters dedicated to these topics while in the other book, these topics are sections within a single chapter. The applied book and the course I took did have some useful "math stuff" like learning about the moment generating functions and how to get the mean, variance, etc. from differentiating said functions (so, fewer things to memorize). It also teaches some useful things like if you have two random variables X and Y, how do you determine the distribution of another random variable, Z, that is a function of X and Y.

 

I guess the "applied" book is like a "toolbox" of probabilistic models and other tricks you can apply to whatever problem you are trying to solve. I don't consider myself an expert in probability theory at all, but instead, I feel that given a data-driven problem and a question, I know what tools to use to answer that question, and where to find those tools in the book. I certainly do not know how to derive/prove these theories myself, and I am not qualified to develop my own statistical methods; but I do know enough "math stuff" to follow along someone else's proof.

Fair enough. My viewpoint is probably because I was a math major. A large part of the reason I forgot so much stuff after studying to pass the SOA probability test is probably that I only had 1-2 weeks to study for it, so I had to cram as much probability into my brain as possible in that time frame.

[TakeruK]

Fair enough. My viewpoint is probably because I was a math major. A large part of the reason I forgot so much stuff after studying to pass the SOA probability test is probably that I only had 1-2 weeks to study for it, so I had to cram as much probability into my brain as possible in that time frame.

 

To be clear, I was not trying to say your approach is bad/less ideal. I was trying to understand the differences and provide information to people reading this thread to help them decide which approach they want to read

[django4321]

Here's a full pdf of Mathematical Statistics w/ Applications 7th ed. by Wackerly, Mendenhall, and Scheaffer:

https://fvela.files.wordpress.com/2012/01/mathematical_statistics_with_applications1.pdf

This is the book my Stats program uses in our three semester Mathematical Statistics series (Probability, Mathematical Statistics, Theory of Statistical Inference).

I think it's a solid book for the most part. If you'd like I could send you some of the practice tests my prof put together for each chapter.

Edited November 14, 2015 by django4321
typo

[StatsG0d]

For the first year sequence in probability, my program uses Casella and Berger's Statistical Inference. The first few chapters are devoted to probability theory, while the rest of the book is primarily mathematical statistics.

[Ellies]

On ‎12‎/‎29‎/‎2015 at 8:23 AM, footballman2399 said:

For the first year sequence in probability, my program uses Casella and Berger's Statistical Inference. The first few chapters are devoted to probability theory, while the rest of the book is primarily mathematical statistics.

Same here; that's what my program uses (Rutgers).