(Part 2) Best probability & statistics books according to redditors
We found 1,070 Reddit comments discussing the best probability & statistics books. We ranked the 426 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
I've wasted too much time trying to find the so-called "right" statistics book. I'm still early in my journey, going through calculus using Prof. Leonards videos while working through a Linear Algebra book all in prep for tackling a stats book. Here's a list of books that I've had a look at so far.
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These seem to be of a similar level with regards to rigour, as they aren't that rigourous. That's not to say you can get by without the calculus prereq and even linear algebra
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The other two I've been looking at which seem to be a lot more complex are
And then there's Casella and Berger's Statistical inference, which I looked at once and decided not to look at again until I can manage at least one of the aforementioned books. I think I'm leaning most to the first book listed. Whichever one you decide to use, good luck with your journey.
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You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.
Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.
Basically, don't limit yourself to the track you see before you. Explore and enjoy.
Analyzing Baseball Data with R is basically the textbook when it comes to learning R with baseball data. Second edition just came out with some updates with regards to Statcast data. It's like $50, though, so if that's a bit out of your reach, I might suggest looking at DataCamp's R tutorials or something of that nature.
CLRS for algorithms/CS.
Probability and random processes for statistics.
Biological Sequence Analysis by Richard Durbin for my subfield of bioinformatics.
It is hard to provide a "comprehensive" view, because there's so much disperate material in so many different fields that draw upon probability theory.
Feller is an approachable classic that covers all of the main results in traditional probability theory. It certainly feels a little dated, but it is full of the deep central limit insights that are rarely explained in full in other texts. Feller is rigorous, but keeps applications at the center of the discussion, and doesn't dwell too much on the measure-theoretical / axiomatic side of things. If you are more interested in the modern mathematical theory of probability, try Probability with Martingales.
On the other hand, if you don't care at all about abstract mathematical insights, and just want to be able to use probabilty theory directly for every-day applications, then I would skip both of the above, and look into Bayesian probabilistic modelling. Try Gelman, et. al..
Of course, there's also machine learning. It draws on a lot of probability theory, but often teaches it in a very different way to a traditional probability class. For a start, there is much more emphasis on multivariate models, so linear algebra is much more central. (Bishop is a good text).
For undergrad probability, Pitman's book or Ross's two books here and here.
For graduate probability, Billingsley (h/t /u/DCI_John_Luther), Williams or Durrett.
I'd like to give you my two cents as well on how to proceed here. If nothing else, this will be a second opinion. If I could redo my physics education, this is how I'd want it done.
If you are truly wanting to learn these fields in depth I cannot stress how important it is to actually work problems out of these books, not just read them. There is a certain understanding that comes from struggling with problems that you just can't get by reading the material. On that note, I would recommend getting the Schaum's outline to whatever subject you are studying if you can find one. They are great books with hundreds of solved problems and sample problems for you to try with the answers in the back. When you get to the point you can't find Schaums anymore, I would recommend getting as many solutions manuals as possible. The problems will get very tough, and it's nice to verify that you did the problem correctly or are on the right track, or even just look over solutions to problems you decide not to try.
Basics
I second Stewart's Calculus cover to cover (except the final chapter on differential equations) and Halliday, Resnick and Walker's Fundamentals of Physics. Not all sections from HRW are necessary, but be sure you have the fundamentals of mechanics, electromagnetism, optics, and thermal physics down at the level of HRW.
Once you're done with this move on to studying differential equations. Many physics theorems are stated in terms of differential equations so really getting the hang of these is key to moving on. Differential equations are often taught as two separate classes, one covering ordinary differential equations and one covering partial differential equations. In my opinion, a good introductory textbook to ODEs is one by Morris Tenenbaum and Harry Pollard. That said, there is another book by V. I. Arnold that I would recommend you get as well. The Arnold book may be a bit more mathematical than you are looking for, but it was written as an introductory text to ODEs and you will have a deeper understanding of ODEs after reading it than your typical introductory textbook. This deeper understanding will be useful if you delve into the nitty-gritty parts of classical mechanics. For partial differential equations I recommend the book by Haberman. It will give you a good understanding of different methods you can use to solve PDEs, and is very much geared towards problem-solving.
From there, I would get a decent book on Linear Algebra. I used the one by Leon. I can't guarantee that it's the best book out there, but I think it will get the job done.
This should cover most of the mathematical training you need to move onto the intermediate level physics textbooks. There will be some things that are missing, but those are usually covered explicitly in the intermediate texts that use them (i.e. the Delta function). Still, if you're looking for a good mathematical reference, my recommendation is Lua. It may be a good idea to go over some basic complex analysis from this book, though it is not necessary to move on.
Intermediate
At this stage you need to do intermediate level classical mechanics, electromagnetism, quantum mechanics, and thermal physics at the very least. For electromagnetism, Griffiths hands down. In my opinion, the best pedagogical book for intermediate classical mechanics is Fowles and Cassidy. Once you've read these two books you will have a much deeper understanding of the stuff you learned in HRW. When you're going through the mechanics book pay particular attention to generalized coordinates and Lagrangians. Those become pretty central later on. There is also a very old book by Robert Becker that I think is great. It's problems are tough, and it goes into concepts that aren't typically covered much in depth in other intermediate mechanics books such as statics. I don't think you'll find a torrent for this, but it is 5 bucks on Amazon. That said, I don't think Becker is necessary. For quantum, I cannot recommend Zettili highly enough. Get this book. Tons of worked out examples. In my opinion, Zettili is the best quantum book out there at this level. Finally for thermal physics I would use Mandl. This book is merely sufficient, but I don't know of a book that I liked better.
This is the bare minimum. However, if you find a particular subject interesting, delve into it at this point. If you want to learn Solid State physics there's Kittel. Want to do more Optics? How about Hecht. General relativity? Even that should be accessible with Schutz. Play around here before moving on. A lot of very fascinating things should be accessible to you, at least to a degree, at this point.
Advanced
Before moving on to physics, it is once again time to take up the mathematics. Pick up Arfken and Weber. It covers a great many topics. However, at times it is not the best pedagogical book so you may need some supplemental material on whatever it is you are studying. I would at least read the sections on coordinate transformations, vector analysis, tensors, complex analysis, Green's functions, and the various special functions. Some of this may be a bit of a review, but there are some things Arfken and Weber go into that I didn't see during my undergraduate education even with the topics that I was reviewing. Hell, it may be a good idea to go through the differential equations material in there as well. Again, you may need some supplemental material while doing this. For special functions, a great little book to go along with this is Lebedev.
Beyond this, I think every physicist at the bare minimum needs to take graduate level quantum mechanics, classical mechanics, electromagnetism, and statistical mechanics. For quantum, I recommend Cohen-Tannoudji. This is a great book. It's easy to understand, has many supplemental sections to help further your understanding, is pretty comprehensive, and has more worked examples than a vast majority of graduate text-books. That said, the problems in this book are LONG. Not horrendously hard, mind you, but they do take a long time.
Unfortunately, Cohen-Tannoudji is the only great graduate-level text I can think of. The textbooks in other subjects just don't measure up in my opinion. When you take Classical mechanics I would get Goldstein as a reference but a better book in my opinion is Jose/Saletan as it takes a geometrical approach to the subject from the very beginning. At some point I also think it's worth going through Arnold's treatise on Classical. It's very mathematical and very difficult, but I think once you make it through you will have as deep an understanding as you could hope for in the subject.
Anti-disclaimer: I do have personal experience with all the below books.
I really enjoyed Lee for Riemannian geometry, which is highly related to the Lorentzian geometry of GR. I've also heard good things about Do Carmo.
It might be advantageous to look at differential topology before differential geometry (though for your goal, it is probably not necessary). I really really liked Guillemin and Pollack. Another book by Lee is also very good.
If you really want to dig into the fundamentals, it might be worthwhile to look at a topology textbook too. Munkres is the standard. I also enjoyed Gamelin and Greene, a Dover book (cheap!). I though that the introduction to the topology of R^n in the beginning of Bartle was good to have gone through first.
I'm concerned that I don't see linear algebra in your course list. There's a saying "Linear algebra is what separates Mathematicians from everyone else" or something like that. Differential geometry is, in large part, about tensor fields on manifolds, and these are studied by looking at them as elements of a vector space, so I'd say that linear algebra is something you should get comfortable with before proceeding. (It's also great to study it before taking quantum.) I can't really recommend a great book from personal experience here; I learned from poor ones :( .
Also, there are physics GR books that contain semi-rigorous introductions to differential geometry, even if these sections are skipped over in the actual class. Carroll is such a book. If you read the introductory chapter and appendices, you'll know a lot. On the differential topology side of things, there's Schutz, which is a great book for breadth but is pretty material dense. Schwarz and Schwarz is a really good higher level intro to special relativity that introduces the mathematical machinery of GR, but sticks to flat spaces.
Finally, once you have reached the mountain top, there's Hawking and Ellis, the ultimate pinnacle of gravity textbooks. This one doesn't really fall under the anti-disclaimer from above; it sits on my shelf to impress people.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
I would start with Cover & Thomas' book, read concurrently with a serious probability book such as Resnick's or Feller's.
I would also take a look at Mackay's book later as it ties notions from Information theory and Inference together.
At this point, you have a grad-student level understanding of the field. I'm not sure what to do to go beyond this level.
For crypto, you should definitely take a look at Goldreich's books:
Foundations Vol 1
Foundations Vol 2
Modern Crypto
Took a numerical statistics course first year of my masters which used C&B as well. I found Hogg, McKean, and Craig - Introduction to Mathematical Statistics to be a really good companion volume. It covers a lot of the same material, but in a more accessible fashion.
You beat me to it! Well, here are the recommendations:
> On advanced Bayesian statistics, Cyan recommends Gelman's Bayesian Data Analysis over Jaynes' Probability Theory: The Logic of Science and Bernardo's Bayesian Theory.
> On basic Bayesian statistics, jsalvatier recommends Skilling & Sivia's Data Analysis: A Bayesian Tutorial over Gelman's Bayesian Data Analysis, Bolstad's Bayesian Statistics, and Robert's The Bayesian Choice.
I remember reading The man who counted about a million years ago in high school. I think it was pretty good (there's a particular "inheritance division" story which is pretty nice and I still remember a bit).
In college I picked up Conway and Guy's Book of numbers and I still think it's one of the best math books ever.
Then you have the obvious Anything Martin Gardner wrote suggestion which cant be bad. If you want to learn "serious math" he has an annotated version of an old book on calculus based on infinitesimals which some people are really into.
A bit meta but I for one enjoyed Polya's books on Mathematics and plausible reasoning (and also the very short but nice "How to solve it").
Also there's the very nice collection of particularly elegant reasoning Proofs from the book but the math is pretty advanced I guess.
A good way to learn math while having fun is to look at problem collections. I remember Halmos' Problems for mathmaticians young and old giving me many nice challenges to think of on the bus.
Those are all pretty old books. A bit newer is a book I haven't read it but I've heard really nice things about: Persi Diaconis and Ron Graham's book Magical mathematics which I think explains the math behind different types of magic tricks.
Hope this helps! Have a good time!
The 1st Course In Math Analysis by Brannan
Analysis I by Terrence Tao
Yet Another Intro To Real Analysis by Bryant
Understanding Analysis Stephen Abbott
Elementary Analysis: The Theory of Calculus Kenneth A. Ross
Metric Spaces by Robert Reisel
A Problem Text in Advanced Calculus by John Erdman. PDF
Advanced Calculus by Shlomo Sternberg and Lynn Loomis.PDF
I've been going through this book. Would recommend.
Andrew Gelman is the one of the authors of Bayesian Data Analysis. He generally favors Bayesian approaches to statistics, although I get the impression he sees them as means to getting robust/tractable and partially-pooled estimates from data, rather than as the only coherent way to make any inferences, ever.
If you're comfortable with Python and some math notation Python Machine Learning, is a great resource for getting started. There's a great balance between explaining concepts and applying code.
In Machine Learning, knowledge of statistics is a huge help. This book explains basic concepts and this Pycon talk applies them in practice.
If you're looking to understand concepts and theories, Calculus and a bit of Linear Algebra will go a long way.
You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:
Real Mathematical Analysis by Charles Pugh
Understanding Analysis by Stephen Abbot
A Primer of Real Functions by Ralph Boas
Yet Another Introduction to Analysis
Elementary Analysis: The Theory of Calculus
Real Analysis: A Constructive Approach
Introduction to Topology and Modern Analysis by George F. Simmons
...and tons more.
In my opinion the hardest part of the course is the first 3 weeks, and the last 3-4 weeks.
FIRST THREE WEEKS:
Probability at first was extremely confusing, and in some ways still is a bit confusing for me since I almost never use it and forget stuff over time. You may be like me in this regard, the reason I always would get tricked by probability is there are cases where the wrong answer just seems like pure common sense (until you learn probability better) which will leads you down a very wrong path because you are convincing yourself you're right when you are not. The trick I found for myself was to aggressively do every problem I could get my hands on and understand exactly why I was wrong. I went through 2-3 different textbooks outside of the course and only then finally started to understand how to think in a probabilistic way whereby the tricks that tend to destroy people on exams and such would not catch me off guard.
The textbook for the course (Grinstead and Snell? I may be spelling this wrong) was extremely verbose and I started reading elsewhere out of boredom, in retrospect I regret this decision since it was the closest book to all the topics covered in 247.
The lecturer felt like he threw examples at us (I assume this is your complaint too?) and my biggest mistake in that course was not spending ample amounts of time understanding exactly why they worked. Despite this, what he did explain was good and I liked his teaching a lot, but I had to go to office hours to understand things that were vague in lectures.
As an example, do you know why the permutation formula is defined the way it is? Do you know why n choose k is defined the way it is, or rather, how can you get to the formula for n choose k if you know the permutation formula?
The unfortunate thing is it took me going through books like A First Course in Probability, which is probably insane to go through if you aren't comfortable with math/proofs/some stats already despite the book name... but the massive amount of examples gave me some pretty huge insights. I did this for I think 1-2 other books, and then I read the textbook for the course. It was not easy, you will invest probably 2x the work of any other class if you try what I did, and I didn't even do as much as I'm telling you here and tried half of this after the course.
The best thing you may be able to do if you're like me is just practice more and more problems, make sure you fully understand exactly why you were wrong if so, and double confirm why you were right just to make sure you didn't arrive on the answer via some fluke -- which I actually had happen to me on the midterm and gave me an over-inflated mark because... luck. You must understand every detail of why the formula exists the way that it does. I say this because the amount of dumb tricks on the midterms will not be pleasant if you get caught up on "am I actually right?" like I do and choke.
Also the fact that each question on the midterm was 1-2% of your mark also caused a great deal of stress, and I don't perform too well under it.
LAST n-1 TO n-4 WEEKS:
I rushed moment generating functions because I fucked up my study time when CSC236 came around for midterms and shallowly understood them as a result. This was a mistake, so don't do this. It's quite cool what you can do with it actually so let that inspire you.
Chapter 9 (or the last few weeks minus the very last week) was double integration with stuff, and I was not only extremely rusty at this but unable to find any external practice whatsoever. I went through the entire lectures having no clue why we were doing it, and due to a severe lack of time I memorized a ton of formulas instead of understanding... and paid the price on the final for that very reason (causing me to drop from an A-/A to a B, which pissed me off tremendously and was all my fault).
The very last unit which was Markov Chains for me was common sense and extremely interesting, and the exam questions were very straightforward with no tricks... or so it seemed...
And that is my experience with the course.
That class average was the lowest out of every course I've ever had, also was my lowest mark too, I wish I spent more time understanding. I found the middle weeks (mainly 3 - 6) to be straight forward and number crunchy with a lot more intuition, but you'll likely still have to haul ass for that section too if its your first time looking at that.
Maybe I would have had better luck in STA257 if they go into deeper understand of why with proofs, I don't know...
Try the two:
https://www.amazon.com/Introduction-Mathematical-Statistics-Robert-Hogg/dp/0321795431
https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
introduction to mathematical statistics by craig and statistical inference by george casella.
Then read Analyzing Baseball Data With R
http://www.amazon.com/First-Course-Probability-9th-Edition/dp/032179477X
http://www.amazon.com/Mathematical-Statistics-Analysis-Available-Enhanced/dp/0534399428
http://www.amazon.com/Probability-Measure-Patrick-Billingsley/dp/1118122372
http://www.amazon.com/Mathematical-Statistics-Selected-Topics-Edition/dp/0132306379
EDIT:
Most likely Rice will be the best book for a comprehensive look at prob/stat, and it is sufficiently technical.
Schaum's probability and statistics was enough for me when I was a physics undergrad.
I'd recommend Conway and Guy's The Book of Numbers and Abbott's Flatland: A Romance of Many Dimensions.
Shankar's book teaches almost everything you need: calculus, vectors, series, complex variables, ODE, linear algebra in only ~300pag.
http://www.amazon.com/Basic-Training-Mathematics-Fitness-Students/dp/0306450364
For more advanced topics check out Arfken.
Uhm. Do you mean this one?
https://www.amazon.com/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220
Everything and More: A Compact History of Infinity
It inspired me to become a mathematician. Maybe not the most profound, but as far as mathematical books, it's still, to this day, the liveliest and most easily accessible book I've ever read.
I found Geometrical methods of mathematical physics by Bernard Schutz to be helpful, though it doesn't have many problems and it doesn't go into much depth on covariant differentiation. But it is good about discussing the modern view of tensors.
I would recommend watching the first half of the International Winter School on Gravity & Light (check the YouTube channel as well ) if you are interested in learning tensor calculus for use in differential geometry for GR.
I learned tensor calculus in bits from several different courses and texts, so I'm not sure what the best ones that are actually dedicated to the subject might be. In any case, I think you'll have a lot of fun learning the subject.
Conway and Guy's The Book of Numbers
Ross is the standard probability book. Its on its 9th edition, so it most likely has few typos (I have the 5th, and that was a solid book). Also, you can probably (get it?) find an older edition for next to nothing.
I really enjoyed Dantzig's, Number. It more explores the development of the numerical system, but I think that is tied into what you are interested in. The book doesnt really get too far into modern notation though.
I learned from Baez & Muniain; Gauge Fields, Knots, and Gravity.
Toward the end of the course, I met Brian Greene at a public talk, and he recommended Schutz; Geometrical Methods of Mathematical Physics.
Khan Academy, free.
If you want problems and answers, I highly recommend the Schaums guides. You’ll need to pick the right one for her level, but basically there are a lot of problems and answers to help understand the issues.
https://www.amazon.ca/Schaums-Outline-Probability-Statistics-4th/dp/007179557X
Found Naked Statistics to be a great casual read.
https://www.amazon.com/dp/1480590185
There's a 2nd edition of this book coming out in December with added chapters on tidyverse and using statcast data
Perhaps this or this.
Feller. I forgot the author. It is a bit more rigorous. I would try to get it from your university or local library first to see if you will need another book to prepare.
http://www.amazon.com/Introduction-Probability-Theory-Applications-Edition/dp/0471257087
Its become a bit more expensive since when I bought it.
http://www.math.uah.edu/stat/index.html
Might not be exactly what you're looking for, but it's free.
You might also want to try Feller's book. I got a used copy of the third edition on ebay for 5 dollars...I wouldn't recommend paying the price they want for a new one. It's also on bookfi if you're into that.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
If you're looking for a thorough and rigorous introduction into probability theory, I'd recommend going with Introduction to Probability Theory and Its Applications Vol.1 and 2 by Feller. Another well recommended book is Probability and Random Processes by Grimmett and Stirzaker (this starts from the get-go with measure theory).
If you're looking for general statistics, then you may want to look at All of Statistics by Wasserman and perhaps Bayesian Data Analysis by Gelman, et al.
Finally, since you're a physicist, you'll probably want to take a look at Monte Carlo methods in particular, such as with Monte Carlo Statistical Methods by Robert and Casella.
Have you had a look at Carroll's general relativity notes? Chapters 2 and 3 are predominantly about developing the mathematics behind GR, and are very good introductions to this. I have a copy of Carroll's book and I can promise you that those chapters are almost unchanged in the book as compared to the lecture notes. This is my main suggestion really, as the notes are freely available, written by an absolute expert and a joy to read. I can't recommend them (and the book really) enough.
Most undergraduate books on general relativity start with a "physics first" type approach, where the underlying material about manifolds and curvature is developed as it is needed. The only problem with this is that it makes seeing the underlying picture for how the material works more difficult. I wouldn't neccessarily say avoid these sort of books (my favourite two of this kind would be Cheng's book and Hartle's.) but be aware that they are probably not what you are looking for if you want a consistent description of the mathematics.
I would also say avoid the harder end of the scale (Wald) till you've at least done your course. Wald is a tough book, and certainly not aimed at people seeing the material for the first time.
Another useful idea would be looking for lecture notes from other universities. As an example, there are some useful notes here from cambridge university. Generally I find doing searches like "general relativity site:.ac.uk filetype:pdf" in google is a good way to get started searching for decent lecture notes from other universities.
If you're willing to dive in a bit more to the mathematics, the riemannian geometry book by DoCarmo is supposed to be excellent, although I've only seen his differential geometry book (which was very good). As a word of warning, this book might assume knowledge of differential geometry from his earlier book. The book you linked by Bishop also looks fine, and there is also the book by Schutz which is supposed to be great and this book by Sternberg which looks pretty good, although quite tough.
Finally, if you would like I have a dropbox folder of collected together material for GR which I could share with you. It's not much, but I've got some decent stuff collected together which could be very helpful. As a qualifier, I had to teach myself GR for my undergrad project, so I know how it feels being on your own with it. Good luck!
The Book of Numbers by John Conway. Its not particularly "real-world" but it is an excellent book.
Evolution of numbers is the "basic story". This is what Einstein said about this book
"This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of mathematical thought from the earliest times to the latest constructions is presented here with admirable consistency and originality and in a wonderfully lively style."
Read Amazon reviews to get more info.
>My first goal is to understand the beauty that is calculus.
There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.
There are some great intros for RA:
Numbers and Functions: Steps to Analysis by Burn
A First Course in Mathematical Analysis by Brannan
Inside Calculus by Exner
Mathematical Analysis and Proof by Stirling
Yet Another Introduction to Analysis by Bryant
Mathematical Analysis: A Straightforward Approach by Binmore
Introduction to Calculus and Classical Analysis by Hijab
Analysis I by Tao
Real Analysis: A Constructive Approach by Bridger
Understanding Analysis by Abbot.
Seriously, there are just too many more of these great intros
But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers
Discrete Mathematics with Applications by Epp
Mathematics: A Discrete Introduction by Scheinerman
(Note: I wrote this elsewhere)
Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:
When you master these concepts, you will see that all hairy, formal definitions boil down to functions, sets, and propositions (with quantifiers). Recursion appears in many interesting structures in computer science, as well as in proofs of theorems (which are themselves propositions).
I don't know of any good online introduction to discrete math, but here are a few book ideas.
I know this is a boring suggestion, but nothing beats the old, venerable Schaum's Outlines for their combination of problems, solutions, and inexpensiveness. If you are just starting, perhaps start with this one, and once you get some exposure to the basics you'll have a better idea of what you might want to pursue next- perhaps the next step would be analysis using a computer instead of by hand.
Lots of us have free YouTube videos on the basics that you can reference if/when you need them as you go. Try me or Kahn Academy, there are many others. Let me know if this idea doesn't fit with what you had in mind, and I can try to point you in a different direction.
Did any of your calc classes include multivariate/vector calculus? E.g. things dealing with double and triple integrals.
If not, take another calc class or two; calculus is very important for statistics. It shouldn't be too hard to pick up the rest of the necessary calc since you've already got a good calc background.
If so, start taking probability and statistics courses in your school's math department if you can. The mathematical way (read: the right way) of understanding probability and statistics is based on probability distributions (like the normal distribution), defined by their probability functions. As such, you can use calculus to obtain a myriad of information from them! For instance, among many other things, within the first one or 2 courses, you'd likely be able to answer at least the Spearman's coefficient question, the Bernoulli process question, and the MLE question.
If you don't have room in your schedule to do the stats course, you could get a textbook and try learning on your own. There are tons of excellent resources. Hogg, Tanis, and Zimmerman is pretty good for an introduction, though I'm sure there's better out there.
I quite like Schutz’s book:
Geometrical Methods of Mathematical Physics
https://www.amazon.co.uk/dp/0521298873/
https://www.amazon.com/Schaums-Outline-Probability-Statistics-4th/dp/007179557X
https://www.amazon.com/Probability-Through-Problems-Problem-Mathematics/dp/038795063X/
https://www.amazon.com/Problems-Probability-Mathematical-Statistics-Functions/dp/0486637174
https://www.amazon.com/Thousand-Exercises-Probability-Geoffrey-Grimmett/dp/0198572212
It's a good idea to abuse the library if still in school.
Not interested, but related to your mention of R, there's a book about analyzing baseball data using R. Haven't checked it out yet but I plan to.
> Abstract Algebra: A First Course D. Saracino, Waveland Press, 2008. ISBN: 9781478610137 2nd Edition
Only found the 1st ed: https://drive.google.com/open?id=0B6gWPriYv4tMak56YlVsZXhqM2c
Just follow the instruction above for the eBook from Amazon if you want the 2nd ed: https://www.amazon.com/Abstract-Algebra-Course-Dan-Saracino/dp/1577665368/ref=sr_1_1?s=books&ie=UTF8&qid=1503439691&sr=1-1&keywords=Abstract+Algebra+Saracino
For math you're going to need to know calculus, differential equations (partial and ordinary), and linear algebra.
For calculus, you're going to start with learning about differentiating and limits and whatnot. Then you're going to learn about integrating and series. Series is going to seem a little useless at first, but make sure you don't just skim it, because it becomes very important for physics. Once you learn integration, and integration techniques, you're going to want to go learn multi-variable calculus and vector calculus. Personally, this was the hardest thing for me to learn and I still have problems with it.
While you're learning calculus you can do some lower level physics. I personally liked Halliday, Resnik, and Walker, but I've also heard Giancoli is good. These will give you the basic, idealized world physics understandings, and not too much calculus is involved. You will go through mechanics, electromagnetism, thermodynamics, and "modern physics". You're going to go through these subjects again, but don't skip this part of the process, as you will need the grounding for later.
So, now you have the first two years of a physics degree done, it's time for the big boy stuff (that is the thing that separates the physicists from the engineers). You could get a differential equations and linear algebra books, and I highly suggest you do, but you could skip that and learn it from a physics reference book. Boaz will teach you the linear and the diffe q's you will need to know, along with almost every other post-calculus class math concept you will need for physics. I've also heard that Arfken, Weber, and Harris is a good reference book, but I have personally never used it, and I dont' know if it teaches linear and diffe q's. These are pretty much must-haves though, as they go through things like fourier series and calculus of variations (and a lot of other techniques), which are extremely important to know for what is about to come to you in the next paragraph.
Now that you have a solid mathematical basis, you can get deeper into what you learned in Halliday, Resnik, and Walker, or Giancoli, or whatever you used to get you basis down. You're going to do mechanics, E&M, Thermodynamis/Statistical Analysis, and quantum mechanics again! (yippee). These books will go way deeper into theses subjects, and need a lot more rigorous math. They take that you already know the lower-division stuff for granted, so they don't really teach those all that much. They're tough, very tough. Obvioulsy there are other texts you can go to, but these are the one I am most familiar with.
A few notes. These are just the core classes, anybody going through a physics program will also do labs, research, programming, astro, chemistry, biology, engineering, advanced math, and/or a variety of different things to supplement their degree. There a very few physicists that I know who took the exact same route/class.
These books all have practice problems. Do them. You don't learn physics by reading, you learn by doing. You don't have to do every problem, but you should do a fair amount. This means the theory questions and the math heavy questions. Your theory means nothing without the math to back it up.
Lastly, physics is very demanding. In my experience, most physics students have to pretty much dedicate almost all their time to the craft. This is with instructors, ta's, and tutors helping us along the way. When I say all their time, I mean up until at least midnight (often later) studying/doing work. I commend you on wanting to self-teach yourself, but if you want to learn physics, get into a classroom at your local junior college and start there (I think you'll need a half year of calculus though before you can start doing physics). Some of the concepts are hard (very hard) to understand properly, and the internet stops being very useful very quickly. Having an expert to guide you helps a lot.
Good luck on your journey!
For people who want a good grasp of Cantor's idea in prose. I heartily recommend the book Everything and More by the incomparable David Foster Wallace.
For anyone interested in a lay explanation of set theory in a challenging (for laymen) but tremendously well written and engaging book, I'd recommend Everything and More by David Foster Wallace. I'm sure it's beneath most mathematicians, but I really loved it.
Rather than list various courses, I'll say this. If you can use all the techniques in this book:
http://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Edition/dp/0123846544/ref=dp_ob_title_bk/185-3957242-1103639
and understand the content of this book:
http://www.amazon.com/Mathematical-Physics-Sadri-Hassani/dp/0387985794/ref=sr_1_1?s=books&ie=UTF8&qid=1335192374&sr=1-1
then you will almost certainly know all they math you'll ever need for advanced undergraduate and general graduate courses. In fact, you'll almost certainly know much more than you'll need.
That's not to say that you should simply study those books - the second one is a gem, but the first is.... polarizing - but they're useful guides of what you ought to know.
Grimmett and Stirzaker.
found it! Apparently they've gone through several editions and added a coauthor since I bought my copy.
My father is a statistician and he is the one who recommended Hogg and Craig when I complaining about Casella and Berger. I spent a summer working my way through Hogg and Craig and then reviewed everything from my classes that previous year as my way for studying for the written quals. I passed so it worked. And then I promptly forgot everything.
This might get poo pood, but I really like some of the schaums outline books.
https://www.amazon.com/Schaums-Outline-Probability-Statistics-4th/dp/007179557X
Why? They are packed full of sample problems and answers, and they tend to provide really concise definitions. I think one of the better ways to understand conditional probability is to see it applied to a range of clear examples.
Also, these books are ridiculously cheap. Tiny investment to make on the off chance you don’t love the format.
I still use this book to quickly brush up on specific concepts at least once a year.
My intro/grad class used Bulmer's book. Its an enjoyable read, easy to follow and answers to odd exercises in the back... and a hell of alot cheaper than current textbooks.
Probability and Random Processes is a wonderful book in probability; but focused on probability to the point that major statistical distributions (chi-squared, T) are merely asides.
Schaum's Outline of Probability and Statistics is a good review with lots of practice problems. Check out the videos on Khan academy too, they really helped me with some of the concepts.
Interdisciplinary connections spring up from generality. You'd be hard pressed to find a spontaneous connection between something like particle phenomenology and an unrelated field.
To illustrate this idea of generality, consider the methods of statistical mechanics, which are so general that they can be used to describe everything from black holes to ferromagnets. However, the methods have also been used to model neural networks and social dynamics (the latter being accurate enough to successfully recreate historical events.)
What makes statistical mechanics more general than other branches? Probably the fact that it's almost more mathematics than physics, specifically a branch of probability theory regarding highly correlated random variables.
With this in mind, perhaps you'd benefit from focusing your attention on the mathematical ideas that drive physics rather than physics itself. Take the calculus of variations which, whilst developed for problems in classical mechanics, has found applications in mathematical optimisation. Another example being brownian motion, the mathematics of which have been generalised to higher dimensions and applied to finance. The mathematics behind relativity is differential geometry, which has been applied to too many fields to list.
I'd recommend having a look at Mathematical Methods for Physicists by Arfken, Weber and Harris for a broad overview of the methods.
It's a parametric model. The parameters of the model are simply the parameters of the distributions he assumes (or the "hyperparameters" if there's some sort of multilevel modelling) over the visible data he feeds into the model (previous years' results). He's fitting using the Stan software (which uses No-U-Turn-Sampling, other reference and another). Once he gets all the posterior probability distributions over the parameters, it's pretty trivial to simulate the model a bunch of times to see the distribution over outcomes.
The advantage of MCMC is that you don't HAVE to calculate the normalization constant (which is hard). Look at the formal derivation of Metropolis-Hastings on wiki. The basic idea is that it relies on a fraction of posterior probability distributions for generating samples from a distribution. Since the normalization constant is present in both numerator and denominator, it cancels out. So you don't need to calculate it directly, and you only need to know the posterior up to a constant of proportionality. And this is generally much easier to do.
If you want a book to look through this stuff, the classic reference is Gelman's Bayesian Data Analysis (and he'll be coming out with a third edition pretty soon).
https://amzn.com/052138835X
/u/istvan_magyary have you tried libgen?
That's the name of the book. Yet Another Introduction to Analysis
PDF WARN: Introduction to Math Stat by Hogg
Not to be confused with Probability and Math Stat by Tannis and Hogg which is a "first semester" course.
Good blend of theory and "talky-ness", good exercises that test your understanding, most should be do-able from just applying the basics.
A Primer of Real Functions - Boas
Dan Saracino wrote a book on abstract algebra and it was the one I used for my intro to abstract algebra course. I honestly think this is one of the best math books out there. I loved it. Great intro book full of great exercises. I’ll link it for you. I’ve gone through Pinter and I think this one is better. Don’t do Dummit and Foote, Aluffi, or Serge Lang.
Abstract Algebra: A First Course https://www.amazon.com/dp/1577665368?ref=ppx_pop_mob_ap_share
Is it Number: The Language of Science by Dantzig? It covers the early history of math pretty well. I strongly recommend it. So does Albert Einstein.
My math stats textbook is Hogg McKean Craig. I don't think the math would be too much for a computation statistics major, but it would give you a great overview if you're interested in that direction.
http://www.amazon.com/Introduction-Mathematical-Statistics-7th-Edition/dp/0321795431
Probability and Random Processes by Grimmett is a good introduction to probability.
Mathematical Statistics by Wackerly is a comprehensive introduction to basic statistics.
Probability and Statistical Inference by Nitis goes into the statistical theory from heavier probability background.
The first two are fairly basic and the last is more involved but probably contains very few applied techniques.
Here are some books I'd recommend.
General Books
These are general books that are more focused on proving things per se. They'll use examples from basic set theory, geometry, and so on.
Topical Books
For learning topically, I'd suggest starting with a topic you're already familiar with or can become easily familiar with, and try to develop more rigor around it. For example, discrete math is a nice playground to learn about proving things because the topic is both deep and approachable by a beginning math student. Similarly, if you've taken AP or IB-level calculus then you'll get a lot of out a more rigorous treatment of calculus.
I have a special place in my hear for Spivak's Calculus, which I think is probably the best introduction out there to math-as-she-is-spoke. I used it for my first-year undergraduate calculus course and realized within the first week that the "math" I learned in high school — which I found tedious and rote — was not really math at all. The folks over at /r/calculusstudygroup are slowly working their way through it if you want to work alongside similarly motivated people.
General Advice
One way to get accustomed to "proof" is to go back to, say, your Algebra II course in high school. Let's take something I'm sure you've memorized inside and out like the quadratic formula. Can you prove it?
I don't even mean derive it, necessarily. It's easy to check that the quadratic formula gives you two roots for the polynomial, but how do you know there aren't other roots? You're told that a quadratic polynomial has at most two distinct roots, a cubic polynomial has a most three, a quartic as most four, and perhaps even told that in general an n^(th) degree polynomial has at most n distinct roots.
But how do you know? How do you know there's not a third root lurking out there somewhere?
To answer this you'll have to develop a deeper understanding of what polynomials really are, how you can manipulate them, how different properties of polynomials are affected by those manipulations, and so on.
Anyways, you can revisit pretty much any topic you want from high school and ask yourself, "But how do I really know?" That way rigor (and proofs) lie. :)
The first one is a logic question. Specifically propositional logic, though if you are interested in logic you could also study basic predicate logic.
The second one could fall into a few overlapping categories, but I might put it into abstract algebra. I found a very good book that I used to self-prepare for a graduate course was this one by Dan Saracino. You may need to go one level more basic depending on your background, in which case i've been told this book is good.
The Math Book. Beautiful pictures on every page, and MOST of the pages are simple enough that a literate six-year-old could understand them. I've never met a person who didn't enjoy flipping through that book.
...so even if your kid doesn't like it, you will ;)
Absolutely. The stress tensor is a (2, 0) tensor (called contravariant in the physicists definition), which means that it takes two vector inputs to produce a real number.
If you input a vector, say e1 (this may be x-hat, the unit vector in the x direction in Cartesian coordinates), it will return a vector which represents the force per unit area in that direction. It actually returns a 1-form (covariant vector), but in the case of the stress tensor, which is a Cartesian tensor, covariant vectors are the same as contravariant vectors, their duals.
This operation is called tensor contraction, where the tensor only acts on one input and returns another tensor of rank (n-1, m-1), or in the case of the stress tensor, it returns a (1, 0) tensor which is just a covariant vector, or in the case of cartesian tensors, it is just a vector (contravariant).
I encourage anybody who is interested in this stuff to read Schutz's Geometrical Methods of Mathematical Physics, as this book describes tensors fully in the newer language (my definition number 2), and does so with applications to physics. Most tensors in physics are taught in the old indices/transformation law language, and can be quite confusing for first timers.
Principles of Statistics (Bulmer) - this is a very nice introduction to probability and statistics. It takes you through the important distributions (binomial, normal, poisson, etc), laws of probability, central limit theorem, etc. And it's like $10 as an eBook or $15 in paperback.
http://www.amazon.com/Principles-Statistics-Dover-Books-Mathematics/dp/0486637603/ref=sr_1_5?ie=UTF8&qid=1463424228&sr=8-5&keywords=statistics
I use [this book]http://www.amazon.com/Statistics-Manual-Edwin-L-Crow/dp/048660599X) as a reference. It's very small and inexpensive (you may have to buy it direct from Dover, though). It won't go through any derivations, but it covers most of the very important, basic, topics. I also have my old textbook on backup as well. Old editions of textbooks are cheaper, and all the information is the same.
Looking through amazon, this one looks pretty good as well, especially for an inexpensive text. A reviewer said it ends with what we just talked about! Any book that lays down the foundations well enough should be fine. See if you can find one that does correlation analysis, since you'll probably use that later.
Getting more advanced than this may depend on your field. If you're in biology, I'd recommend a book on designing and analyzing scientific experiments. I can't recommend a good title, though, because I'm only familiar with computer experiments (which tend to be easier).
I hope this helps!
Number: The Language of Science
Changed my thinking about math/science to be more compatible with my artistic/philosophical leanings.
Everything and More by David Foster Wallace. Just got back into calculus, mostly so I can have at least a literate understanding of it. Not sure how math majors feel about it.
I would suggest you take a look at the following book, https://www.amazon.com/Analyzing-Baseball-Data-Second-Chapman/dp/0815353510, or the Stanford sports stats class, https://web.stanford.edu/class/stats50/references.html
An oldie but good is Introductory Probability and Statistical Applications by Meyer! I've used newer, more fashionable textbooks (Ross, Miller & Miller) but this one is my favorite for the introductory level. It feels a bit dated at times (e.g. "although we cannot expect all readers to own a personal computer"), but the relevant math hasn't changed much in the passed few decades. It is very clear with more exposition than I've found in the newer books I mentioned.
As for an advanced text, I've heard good things about Probability and Random Processes by Grimmet and Stirzaker. My friends used it in a graduate course on basic probability, and compare it favorably to their undergraduate experiences with probability.
What do you mean by advanced Calculus? Multivariate Calculus without proofs?
Anyway,
Mathematical Analysis and Proof by Stirling
A First Course in Mathematical Analysis by Brannan
Yet Another Introduction to Analysis by Bryant
I'd recommend Discrete Mathematics, Elementary and Beyond By Lovász, Pelikán, and Vesztergombi. It's the book I'm using in my undergraduate discrete math course, and I think it's a great introductory book that explores many areas of discrete math, and should allow you to see which field interests you most.
I looked at similar (WA resident also) but there's only a few community college classes that are interesting (linear algebra, probability, ODE) so then you're looking at UW/WSU tuition. There's a couple applied tracks you could consider: machine learning and financial math:
https://metacademy.org/roadmaps/
http://www.deeplearningweekly.com/pages/open_source_deep_learning_curriculum
https://www.quantstart.com/articles/Quantitative-Finance-Reading-List
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Self study: math for physics texts like Arfken/Harris/Weber, Boas, Riley/Hobson, Thomas Garrity
http://www.goldbart.gatech.edu/PostScript/MS_PG_book/bookmaster.pdf
https://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Comprehensive/dp/0123846544
This imo is a good book for basic probability and mathematical statistics. Super easy read with a lot of examples. [You also mentioned pdf's for books and someone told you library gensis. I can promise this one is on there :)]
For math there isn't much better undergraduate/beginning graduate review than Arfken, Weber, Harris. This will cover most mathematics you'll encounter in your first and maybe second years of graduate studies. Personally I'm not a huge fan of the complex contour integration sections you'll encounter in that book - I much prefer Ahlfors or Rudin for something more on the pure side or Churchill for something more on the applied side of complex analysis. The other sections are, in my opinion, stellar - although I have only the third edition in my possession.
You might find a book like Naked Statistics (https://www.amazon.com/Naked-Statistics-Stripping-Dread-Data/dp/1480590185) pretty helpful. The author uses a lot of common-place terminology and situations and helps the reader develop an intuition for the main ideas in statistics.
Imagine two buses... one is full of marathon runners and the other is full of participants in a festival of sausage. You stop each bus and weigh all the passengers. Since marathon runners tend to be lean and more uniform in size, more of them will be closer to the average weight. People attending a festival of sausage will be more diverse. Some will be thin others chunky, and others quite obese. Each individual's weight is more likely to be farther away from their average weight. In this case, the bus with marathon runners will have a lower variance in weight than the bus with the festival of sausage attendees.
The book does a better job than my paraphrased example.
Personally I make a distinction between scripting and programming that doesn't really exist but highlights the differences I guess. I consider myself to be scripting if I am connecting programs together by manipulating input and output data. There is lots of regular expression pain and trial-and-error involved in this and I have hated it since my first day of research when I had to write a perl script to extract the energies from thousands of gaussian runs. I appreciate it, but I despise it in equal measure. Programming I love, and I consider this to be implementing a solution to a physical problem in a stricter language and trying to optimise the solution. I've done a lot of this in fortran and java (I much prefer java after a steep learning curve from procedural to OOP). I love the initial math and understanding, the planning, the implementing and seeing the results. Debugging is as much of a pain as scripting, but I've found the more code I write the less stupid mistakes I make and I know what to look for given certain error messages. If I could just do scientific programming I would, but sadly that's not realistic. When you get to do it it's great though.
The maths for comp chem is very similar to the maths used by all the physical sciences and engineering. My go to reference is Arfken but there are others out there. The table of contents at least will give you a good idea of appropriate topics. Your university library will definitely have a selection of lower-level books with more detail that you can build from. I find for learning maths it's best to get every book available and decide which one suits you best. It can be very personal and when you find a book by someone who thinks about the concepts similarly to you it is so much easier.
For learning programming, there are usually tutorials online that will suffice. I have used O'Reilly books with good results. I'd recommend that you follow the tutorials as if you need all of the functionality, even when you know you won't. Otherwise you get holes in your knowledge that can be hard to close later on. It is good supplementary exercise to find a method in a comp chem book, then try to implement it (using google when you get stuck). My favourite algorithms book is Numerical Recipes - there are older fortran versions out there too. It contains a huge amount of detailed practical information and is geared directly at computational science. It has good explanations of math concepts too.
For the actual chemistry, I learned a lot from Jensen's book and Leach's book. I have heard good things about this one too, but I think it's more advanced. For Quantum, there is always Szabo & Ostlund which has code you can refer to, as well as Levine. I am slightly divorced from the QM side of things so I don't have many other recommendations in that area. For statistical mechanics it starts and ends with McQuarrie for me. I have not had to understand much of it in my career so far though. I can also recommend the Oxford Primers series. They're cheap and make solid introductions/refreshers. I saw in another comment you are interested potentially in enzymology. If so, you could try Warshel's book which has more code and implementation exercises but is as difficult as the man himself.
Jensen comes closest to a detailed, general introduction from the books I've spent time with. Maybe focus on that first. I could go on for pages and pages about how I'd approach learning if I was back at undergrad so feel free to ask if you have any more questions.
Out of curiosity, is it DLPOLY that's irritating you so much?
Gallian is, in my opinion, one of the worst books ever written. The theorems and proofs are messy, the "applications" are just grasping at straws and the expository is horrid. Don't even get me started on how badly written his proofs of the Sylow Theorems are. You would learn more algebra by reading Twilight.
Yes, D&F is for people with a little bit of mathematical maturity. For someone at this guy's level, I'd recommend Saracino. The first chapter goes over all the basic mathematical techniques you'll need, while assuming nothing about your background, the exposition, problems and examples are good and clear. Saracino is essentially the Baby Rudin of Algebra, accessible to anyone familiar with college algebra and with an interest in math.
For probability I'd recommend Introduction to Probability Theory by Hoel, Port & Stone. It has the best explanations of any probability book I've seen, great examples, and answers to most of the problems are in the back (making it well-suited for self-study). I think it's still the best introductory book on the subject, despite its age. Amazon has used copies for cheap.
For statistics, you have to be more precise as to what you mean by an "average undergraduate statistics" course. There's a difference between the typical "elementary statistics" course and the typical "mathematical statistics" course. The former requires no calculus, but goes into more detail about various statistical procedures and tests for practical uses, while the latter requires calculus and deals more with theory than practice. Learning both wouldn't be a bad idea. For elementary stats there are lots of badly written books, but there is one jewel: Statistics by Freedman, Pisani & Purves. For mathematical statistics, Introduction to Mathematical Statistics by Hogg & Craig is decent, though a bit dry. I don't think that Statistical Inference by Casella & Berger is really any better. Those are the two most-used textbooks on the subject.
The Math Book by Clifford Pickover is kind of a coffeetable style book, I love coming home from work and flipping to a random page as a starting point to exploring whatever concept is discussed on it. The book traces chronologically through 250 great discoveries in mathematics.
I think it's this: https://www.amazon.com/Introduction-Mathematical-Statistics-Robert-Hogg/dp/0321795431/ref=mt_hardcover?_encoding=UTF8&me=
But really, if I remember right, they "use" it the same way we "used" the textbook in 400. (I do like both books though.)
Ah, this is great! Thank you :)
I didn't manage to find the book I need for MATH 3215 though. Is there any way you could get Probability and Statistical Inference, Ninth Edition by Hogg, Tanis and Zimmerman?
A First Course in Probability Theory by Sheldon Ross is the book that was used in my undergrad class. The book is currently on the 9th edition, but you can pick up a copy of the 7th edition in like new condition for under $15 plus shipping.
This is also one of the books that is suggested by the Society of Actuaries for the Probability (P) exam.
When I was that age, I used to really enjoy reading books by Clifford Pickover. He has many books, and there are many reviews on Amazon that can aid you in choosing one that you think would be good for your brother-in-law.
Side note: I had not read a Pickover in a long time, but I recently stumbled across The Math Book, which I purchased and quite enjoy. While it may not be the best option for inspiring a 13-year-old, members of Mathit may find it interesting, so I recommend checking it out!
If you want theoretical / mathematical I would suggest reading a few math, stats or engineering books.
Dover is a great place to find some cheaper reading material. They republish old scientific and math texts that were popular in their time in a smaller sized paperback. They're a nice size to bring around with you and they don't cost much.
Math and stats findings of today build on this knowledge, and much of it is still used in state-of-the-art applications. Or, that math/stats is used as part of some state-of-the-art algorithm. Lots of the newest ML algorithms are blending math from a variety of areas.
Statistical analysis of experimental data
Principals of Statistics
Information Theory
Statistics Manual
Some theory of sampling
Numerical Methods for Scientists and Engineers (Hamming)
Mathematical Handbook for Scientists Engineers
Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables
==
There is also the Data-Science Humble Bundle for more technical / practical skill building.
Discrete mathematics and any proof based math in general is what college based math should be like- if you continue to take upper level math and CS courses, you will undoubtedly face this style of math again. Plug and chug (which is what a lot of calculus is) will no longer be the norm.
There is often a very large learning curve for students who are not used to seeing this type of math- so don't stress out too much about it. Eventually, you'll break a point where everything will make (sort of) sense. I went through the exact same thing when I took discrete for the first time, and I felt like I was getting destroyed on everything (I still suck at some topics) until I suddenly hit a point of clarity where I could see how most topics were tied in together. Mathematics, and especially an introductory discrete course, is cruel in that way- that every topic you learn is inherently related to each other, so if you already fall behind just a little, the mountain to catch up just becomes incredibly massive incredibly fast- and it's hard to even pinpoint a place to even start to catch up.
You may be lost in learning elementary proof techniques, or number theory, and then the next topic (say it's graph theory) utilizes a bunch concepts and previous proofs from number theory, and then the next topic might use something proved in graph theory and number theory, and so on. All of a sudden, nothing makes sense, and to learn topic ___, you need to know graph theory, but to know graph theory, you need to know number theory, but you don't know number theory that well, and some topics in number theory can perhaps be explained by another topic in graph theory (or any topic for that matter) The chain is all interlinked and it may difficult to even see where to start- but it is for this reason that once you cross this steep barrier, most things will suddenly become clear to you.
So I'd advise you to just continue visiting professor office hours, asking more questions, asking for other students' help, doing more and more practice. It may seem like you're getting nowhere, but you're essentially learning a new language right now, so it'll obviously take sometime until you feel as if you know what you're doing. Figuring out where people get the intuition to suggest seemingly random functions or a set of numbers or some assumption will come to you slowly, and slowly you'll break more and more of this chain.
https://www.amazon.com/Discrete-Mathematics-Laszlo-Lovasz/dp/0387955852 is another book my professor enjoyed using as a supplmenet.
There is a very interesting book called Number which explores this a bit. It goes through the history of the development of the concept of what a number is, including the shift to writing math with symbols instead of words.
I can't recommend this book highly enough.
This is really good. But you should be aware of this might be outdated. The basics are the same. I browsed someone has recommended Andrew Ng's course. I've been doing this for my master's degree. You definitely need to know math and statistics. For statistics, I recommend you check this out :https://www.openintro.org/stat/textbook.php
they have high school, university, maybe middle school. Anyway, so many people recommend you lots of things, but I don't think most of them consider you are 13 years old (No offence ). Some knowledge in math are a bit difficult for you to understand at the moment, but don't worry; you can remember it first and try to find some introductionary book. such as https://www.amazon.com/Naked-Statistics-Stripping-Dread-Data/dp/1480590185 it's a good read anyway.
What about Number: The Language of Science by Tobias Dantzig? I came across it at a Border's closing and rather enjoyed it.
http://www.amazon.com/Number-Language-Science-Tobias-Dantzig/dp/0452288118/ref=sr_1_1?ie=UTF8&qid=1347469608&sr=8-1&keywords=Number
Dude, Feller
http://www.amazon.com/Introduction-Probability-Theory-Applications-Vol/dp/0471257087
I believe those were the books used during the 2016-2017 school year (thats when I took discrete II)
From what I understand now, the newest renditions of the course use
Discrete Mathematics and Its Applications by K. Rosen
and
A First Course in Probability by Ross
But it'll depend entirely on who it is that's offering the course during the summer and what they include on their syllabus so I'd wait until seeing what they say to purchase either of the books.
The first book you listed (Mathematics for Computer science) is available for free for anyone to use here
The second is available for free on the Rutgers libraries website so I'd advise you not waste your money buying either of those two.
Hope this helps
Since you're an applied math PhD, maybe the following are good. They are not applied though.
This is the book for first year statistics grad students at OSU.
http://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126/ref=sr_1_1?ie=UTF8&qid=1368662972&sr=8-1&keywords=casella+berger
But, I like Hogg/Craig much more.
http://www.amazon.com/Introduction-Mathematical-Statistics-7th-Edition/dp/0321795431/ref=pd_sim_b_2
I believe each can be found in international editions, and for download on the interwebs.
this book really helped me in undergrad. Has a lot of really good concepts. It went along with a course but it does a great job on its own explaining some of the most relevant concepts to computer science.
This one?
I can't remember, honestly. That's how bad it was. I'll dig out my notes tomorrow.
Prob Theory and Math Stats together was basically this book.
Buy this book http://www.amazon.com/exec/obidos/ASIN/1402757964/cliffordpickover
Try look at numbers from an aesthetic point of view besides just logical numbers.
Number theory and such will help because it opens a world of mystery and you begin to realize just how important numbers really are.
http://en.wikipedia.org/wiki/Number_theory
Geometry is also really cool http://mathworld.wolfram.com/topics/Geometry.html
With computers math has become very visual which allows for a greater experience, when you come across a mathematical concept try and Google an image for it, for instance Google "prime number visualization" hit "image search" and your off to the races with incredible stuff.
My favorite introductory discrete math textbook is https://www.amzn.com/0387955852. (It also appears to be available for less unreasonable prices.)