(Part 3) Reddit mentions: The best probability & statistics books

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.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

Just FYI, because this is not always made clear to people when talking about learning or transitioning to data science: this would be a massive undertaking for someone without a strong technical background.

You've got to learn some math, some statistics, how to write code, some machine learning, etc. Each of those is a big undertaking in itself. I am a person who is completely willing to spend 12 hours at a time sitting at a computer writing code... and it still took me a long time to learn how not to write awful code, to learn the tools around programming, etc.

I would strongly consider why you want to do this yourself rather than hire someone, and whether it's likely you'll be productive at this stuff in any reasonable time frame.

That said, if you still want to give this a try, I will answer your questions. For context: I am not (yet) employed as a data scientist. I am a mathematician who is in the process of leaving academia to become a data science in industry.


> Given the above, what do I begin learning to advance my role?

Learn to program in Python. (Python 3. Please do not start writing Python 2.) I wish I could recommend an introduction for you, but it's been a very long time since I learned Python.

Learn about Numpy and Scipy.

Learn some basic statistics. This book is acceptable. As you're reading the book, make sure you know how to calculate the various estimates and intervals and so on using Python (with Numpy and Scipy).

Learn some applied machine learning with Python, maybe from this book (which I've looked at some but not read thoroughly).

That will give you enough that it's possible you could do something useful. Ideally you would then go back and learn calculus and linear algebra and then learn about statistics and machine learning again from a more sophisticated perspective.

> What programming language do I start learning?

Learn Python. It's a general purpose programming language (so you can use it for lots of stuff other than data), it's easy to read, it's got lots of powerful data libraries for data, and a big community of data scientists use it.

> What are the benefits to learning the programming languages associated with so-called 'data science'? How does learning any of this specifically help me?

If you want a computer to help you analyze data, and someone else hasn't created a program that does exactly what you want, you have to tell the computer exactly what you want it to do. That's what a programming language is for. Generally the languages associated with data science are not magically suited for data science: they just happen to have developed communities around them that have written a lot of libraries that are helpful to data scientists (R could be seen as an exception, but IMO, it's not). Python is not intrinsically the perfect language for data science (frankly, as far as the language itself, I ambivalent about it), but people have written very useful Python libraries like Numpy and scikit-learn. And having a big community is also a real asset.

> What tools / platforms / etc can I get my hands on right now at a free or low cost that I can start tinkering with the huge data sets I have access to now? (i.e. code editors? no idea...)

Python along with libraries like Numpy, Pandas, scikit-learn, and Scipy. This stuff is free; there's probably nothing you should be paying for. You'll have to make your own decision regarding an editor. I use Emacs with evil-mode. This is probably not the right choice for you, but I don't know what would be.


> Without having to spend $20k on an entire graduate degree (I have way too much debt to go back to school. My best bet is to stay working and learn what I can), what paths or sequence of courses should I start taking? Links appreciated.

I personally don't know about courses because I don't like them. I like textbooks and doing things myself and talking to people.

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!

I'm from a social science background and, like you, I often find myself hopelessly lost when it comes to what feels like very basic concepts in statistics. I think that's partly due to how statistics is taught in all non-mathematics disciplines - in theory we're taught how to use and evaluate quite complex statistical procedures, but with only 1-2 hours per week teaching, it's impossible for our lecturers to cover the fundamental building blocks that help us to understand what's actually going on.

Because of this, I've recently started a few MOOCs on Coursera, and I've found these massively helpful for covering research methods and statistics in far more depth than my undergraduate and postgraduate lecturers ever had time to delve into. In particular, a couple of courses I'd recommend are:

• Methods and Statistics in Social Sciences - This is particularly focused on quantitative methods in the social sciences (including quite a bit on behavioural and self-report research) so I'm not sure if it will be directly relevant with respect to neuroimaging and cognitive neuroscience, but this gives a great introduction to research methods in general. I've actually only done the first course in this series (Quantitative Research Methods), but they're very comprehensive and well made, so I'm confident that the whole series will be useful for any researcher.
  
• Probability and statistics: To p or not to p? - This one is a little bit more maths-heavy so might be a bit intimidating if you don't find that sort of material easy, but it's a good introduction to some of the core concepts in quantitative research, including some you mentioned (e.g. probability distributions). You don't really have to fully engage with or grasp the maths for it to be useful either.
  
  In terms of textbooks, I personally use Andy Field's Discovering Statistics Using R, and find that very helpful. Field is a psychologist who is very open about his difficulties with learning statistics, and I've found it quite useful and re-assuring to learn from someone with that mindset. He's also tried writing a statistics textbook in the form of a graphic novel, An Adventure in Statistics: The Reality Enigma, so if that sounds like something that might help you, check it out.
  
  I think a few people from a 'purer' statistics background are a bit more critical about Field's books because they're not as comprehensive as a book written by, for example, a statistics professor - and there might be some advice in there that's a little bit out-of-date or not quite correct. He also has a very hit-and-miss cheesy sense of humour, which you'll either love or find very annoying. But I think he takes the right sort of approach for helping people who aren't necessarily mathematically-inclined to dip their toes into the world of statistics.

If you do better with physical materials than online reading, it sounds like renting a textbook would be the way to go for you? Shop around for yourself, but something like this might be good since it covers Algebra 2 and Trig, and renting instead of buying usually only costs around $20. Just remember that if you get stuck or need help there's a lot of resources online too.

Would online lecture videos potentially be helpful for you? Since they're audio visual rather than written, I'm curious if that might be better than learning through reading. Professor Leonard is a personal favorite of mine, although I have to adjust the speed to x1.5 because I feel like he talks really slowly. He has lectures for Algebra, Stats, Precalc, all the way up to Multi Variable Calculus.

As for Kahn Academy, it's definitely more than a lesson supplement, if you wanted to you could learn of all of math from the basics up to college level without having to step foot in a classroom. I personally enjoy it because the online practice problems are much more fun to me than written homework, since you get instant feedback if you get an answer wrong, and help to see how to correct it.

I'll acknowledge that some of the practice problems on Kahn Academy don't get as rigorously difficult as what you might come across in a textbook, as the goal is more to make sure you understand the basic concepts than to test the limit of your problem solving skills.

If you did want to give Kahn a try, then I'd recommend to start from the beginning in Algebra. It should be easy initially since you already completed Algebra 1, but it will also serve as a good review. On the KA home page click on "Subjects" in the upper left corner and select "Algebra." Near the top it should say "Explore" "Classes" "Practice" and "Mission".

Select "Practice," and then start with the first problems and work your way through. If it's too easy initially, then scroll down to the section titled "Functions," as this will be closer to where you left off in Algebra 1. And like I said, you can probably skip the video lessons, and just use the "hint" button, but the relevant videos are also linked with every practice problem if you do get stuck or don't understand something.

> Or that communism creates starvation (joke)

I don't think this is a joke. While causal designs would be difficult to apply, the spatio-temporal correlation is hard to ignore.


>Regarding causality- as you know that’s nearly impossible to prove in the social sciences.

Actually, these days the application of designs and approaches that provide strong support for causal claims have become quite prevalent. Some standard references-



1

2

3

4

good framework reference or a slightly heavier read

and the old classic


In fact, the Nobel prize in economics this year went to some people who have built their careers doing exactly that

It's actually become quite hard to publish in ranking journals in some fields without a convincing (causal) identification strategy.


But we digress.


>We will never be able to do an apples to apples study between heterosexual and homosexual child rearing for some of the reasons you mentioned above. (Diversity of relationship styles, not both biological parents within gay/lesbian couples)

In this case it isn't far fetched at all. The data collection for the survey data used in the study you linked could just as easily have disagregated the parents involved in same sex romantic relationships instead of pooling them. If I understood correctly, the researcher had obtained the data as a secondary source, so they didn't have control over this.

Outcomes for children in the foster care system are well studied, so one could in principal easily replicate the study comparing outcomes between children in the foster care system and those adopted into homes shared by stable same sex couples (you couldn't likely restrict it to married same sex couples, though, because laws permitting same sex civil marriage are too recent to observe outcomes).

>My bottom line-that I don’t see many disagree with if they are being intellectually honest, is a stable monogamous heterosexual family structure is the best model for immediate families. Or would you disagree?

But that's not the question at hand, is it? What we are interested in here is comparing kids bouncing around the state care system to those adopted into homes with two same-sex parents in a stable relationship.

That is exactly my point. The comparison you propose is uninformative relative to the question of permitting same sex couples to "foster to adopt". Because the counterfactual for those children is not likely to be a "stable monogamous heterosexual family". It is bouncing around the foster care system.

> Is your claim that, as you put it, "causal" algebraic equations are in fact not algebraic, and do not conform to the rules of algebra? Can you point me to your source of this new math?

I'm sorry, but I cannot stress this enough. It wasn't ME who developed the use of math to describe causal relationships. After reading the linked wikipedia pages, you're going to have to take that up with the relevant mathematicians/engineers/scientists, etc.

> Can you point me to your source of this new math?

SURE THING. Here's another wikipedia page. But just to ask, I'm curious why you want to use the word "new" for this, given that the wikipedia page cites the standard textbook written in the year 2000, (16 years ago). In fact, by now the concept is so old and established that there are textbooks of it

> The lunacy in your statements is highlighted by the fact that algebraic equations are bidirectional "causal", this is the point of the equal sign. Please read about the equal sign.

See the wikipedia page for causality and then get back to me. Here is a relevant piece of text:

• Nobel Prize laureate Herbert A. Simon and philosopher Nicholas Rescher claim that the asymmetry of the causal relation is unrelated to the asymmetry of any mode of implication that contraposes. Rather, a causal relation is not a relation between values of variables, but a function of one variable (the cause) on to another (the effect). So, given a system of equations, and a set of variables appearing in these equations, we can introduce an asymmetric relation among individual equations and variables that corresponds perfectly to our commonsense notion of a causal ordering.
  
  I highlighted the author's name. That way there should be no confusion about WHO published WHAT (hint: I wasn't me, I'm just a guy who learned how to use causal modeling in college, and who occasionally uses it on the job.).
  
  If using math to model causality is something you personally do not believe has any merit (for whatever personal reasons), that is your own prerogative. I get that you want to pretend this specific use of math doesn't real, in the same way that you said earlier that traders using models for trading doesn't real (I guess it means that in our opinion, the use of proprietary trading models is ALSO not a thing. right?). But yeah, modeling causal relationship is a real thing. And you can rant about it on the internet all day long I suppose, but if you ever go to the doctor, you might find out that causal modeling is used not just in finance, but also in medical sciences.
  
  
  > The lunacy in your statements
  
  DO you mean the part where I cite the research and publication of others? Did you write to any of those authors yet to tell to share any sort of evidence which can challenge anything they've published? I encourage you to read up. So far this conversation is all ME sharing sources and you shouting "doesn't real" (while not being able to produce any sources whatsoever to reinforce that view).
  
  

I have a few books I read at that age that were great. Most of them are quite difficult, and I certainly couldn't read them all to the end but they are mostly written for a non-professional. I'll talk a little more on this for each in turn. I also read these before my university interview, and they were a great help to be able to talk about the subject outside the scope of my education thus far and show my enthusiasm for Maths.

Fearless Symmetry - Ash and Gross. This is generally about Galois theory and Algebraic Number Theory, but it works up from the ground expecting near nothing from the reader. It explains groups, fields, equations and varieties, quadratic reciprocity, Galois theory and more.

Euler's Gem - Richeson This covers some basic topology and geometry. The titular "Gem" is V-E+F = 2 for the platonic solids, but goes on to explain the Euler characteristic and some other interesting topological ideas.

Elliptic Tales - Ash and Gross. This is about eliptic curves, and Algebraic number theory. It also expects a similar level of knowlege, so builds up everything it needs to explain the content, which does get to a very high level. It covers topics like projective geometry, algebraic curves, and gets on to explaining the Birch and Swinnerton-Dyer conjecture.

Abel's proof - Presic. Another about Galois theory, but more focusing on the life and work of Abel, a contemporary of Galois.

Gamma - Havil. About a lesser known constant, the limit of n to infinity of the harmonic series up to n minus the logarithm of n. Crops up in a lot of places.

The Irrationals - Havil. This takes a conversational style in an overview of the irrational numbers both abstractly and in a historical context.

An Imaginary Tale: The Story of i - Nahin. Another conversational styled book but this time about the square root of -1. It explains quite well their construction, and how they are as "real" as the real numbers.

Some of these are difficult, and when I was reading them at 17 I don't think I finished any of them. But I did learn a lot, and it definitely influenced my choice of courses during my degree. (Just today, I was in a two lectures on Algebraic Number Theory and one on Algebraic Curves, and last term I did a lecture course on Galois Theory, and another on Topology and Groups!)

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.

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:

1. Understanding Analysis by Steve Abbot
   
   
2. Yet Another Introduction to Analysis by Victor Bryant
   
   
3. Elementary Analysis: The Theory of Calculus by Kenneth Ross
   
   
4. Real Mathematical Analysis by Charles Pugh
   
   
5. A Primer of Real Functions by Ralph Boas
   
   
6. A Radical Approach to Real Analysis by David Bressoud
   
   
7. The Way of Analysis by Robert Strichartz
   
   
8. Foundations of Analysis by Edmund Landau
   
   
9. A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamzi
   
   
10. Calculus by Spivak
    
    
11. Real Analysis: A Constructive Approach by Mark Bridger
    
    
12. Differential and Integral Calculus by Richard Courant, Edward McShane, Sam Sloan and Marvin Greenberg
    
    
13. You can find tons more if you search the internet. There are more superstars of advanced Calculus like Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by Tom Apostol, Advanced Calculus by Shlomo Sternberg and Lynn Loomis... there are also more down to earth titles like Limits, Limits Everywhere:The Tools of Mathematical Analysis by david Appelbaum, Analysis: A Gateway to Understanding Mathematics by Sean Dineen...I just dont have time to list them all.
    
    Some user friendly books on Linear/Abstract Algebra:
    
    
14. A Book of Abstract Algebra by Charles Pinter
    
    
15. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl Meyer
    
    
16. Groups and Their Graphs by Israel Grossman and Wilhelm Magnus
    
    
17. Linear Algebra Done Wrong by Sergei Treil-FREE
    
    
18. Elements of Algebra: Geometry, Numbers, Equations by John Stilwell
    
    Topology(even high school students can manage the first two titles):
    
    
19. Intuitive Topology by V.V. Prasolov
    
    
20. First Concepts of Topology by William G. Chinn, N. E. Steenrod and George H. Buehler
    
    
21. Topology Without Tears by Sydney Morris- FREE
    
    
22. Elementary Topology by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and and V. M. Kharlamov
    
    Some transitional books:
    
    
23. Tools of the Trade by Paul Sally
    
    
24. A Concise Introduction to Pure Mathematics by Martin Liebeck
    
    
25. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston
    
    
26. Introductory Mathematics: Algebra and Analysis by Geoffrey Smith
    
    
27. Elements of Logic via Numbers and Sets by D.L Johnson
    
    Plus many more- just scour your local library and the internet.
    
    Good Luck, Dude/Dudette.

John Fox's book is great. It's mostly linear regression models for continuous variables, but the GLM section is very helpful. If I remember correctly, the second edition is way more helpful with GLM than the first.

For categorical variables Scott Long's book is wonderfully helpful.

Unfortunately both are expensive. Hopefully your library has them.

Any more specificity in what types of variables you might be working with or what your data is like? Knowing what type of link function you're looking for my give you better results from some of the uber statisticians here.

It would be pretty hard to overstate the importance of the notion of symmetry in physics. Symmetry is explained via algebra (group theory in particular) and geometry. And so studying algebra ("abstract algebra") is going to be a very important thing if you want to continue to understand physics at a higher level .

I would strongly recommend taking an undergraduate math course in abstract algebra (which will always include basic group theory) and depending upon how far you go maybe even a graduate level one.

There are also lots of books aimed at physics students that discuss symmetry and group theory and some basic ideas in topology and geometry that are easy to find.

When I was in school it was popular for physics students to do independent studies with a physics professor in group theory and its applications to physics. I took a reading course like that and read most of this book for it. It was definitely one of the most fun things I did in school. It also gave me a lot more perspective when I started taking graduate level physics courses


Group Theory and Physics https://www.amazon.com/dp/0521558859/ref=cm_sw_r_cp_apa_i_gTLsDbXRB8NZ7

>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

Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.

Aside from that, try these:

Excursions In Calculus by Robert Young.

Calculus:A Liberal Art by William McGowen Priestley.

Calculus for the Ambitious by T. W. KORNER.

Calculus: Concepts and Methods by Ken Binmore and Joan Davies

You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:

[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1413311074&sr=1-1&keywords=spivak+calculus).

Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:

Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.

Analytic Inequalities by Nicholas D. Kazarinoff.

As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.

My plan would go like this:

1. Learn the geometry of LA and how to prove things in LA:

Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.

Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
by Richard Bronson and Gabriel B. Costa.

2. Getting a bit more sophisticated:

Linear Algebra Done Right by Sheldon Axler.

Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.

Linear Algebra Done Wrong by Sergei Treil.

3. Turn into the LinAl's 1% :)

Advanced Linear Algebra by Steven Roman.

Good Luck.

That is a pretty big red flag. Most departments offer a statistics course for non-math majors, I've TA'd that course before, its not good for people who enjoy math. Make sure there is at the very least a calculus pre-req for the course, and you should take a probability course first anyways.

Probability and Statistics in general is such a great field, it would be really unfortunate if a bad class designed for psych majors turned you off from the subject. I would wait until you can take the right classes so you can at least see the material presented in the right way, if your curious what a course sequence should look like:

1. Calculus based probability (undergrad, math major prob)
   
   
2. Calculus based statistics (undergrad, math major stat)
   
   
3. Measure-theoretic probability (grad prob)
   
   
4. Statistical Theory (grad stat - here is where analysis and some really neat math merges with the stats, but undergrad stat does this to some degree as well.
   
   Here are links to textbooks in the same order:
   
   
5. http://www.amazon.com/First-Course-Probability-9th-Edition/dp/032179477X
   
   
6. http://www.amazon.com/Mathematical-Statistics-Analysis-Available-Enhanced/dp/0534399428
   
   
7. http://www.amazon.com/Probability-Measure-Patrick-Billingsley/dp/1118122372
   
   
8. http://www.amazon.com/Mathematical-Statistics-Selected-Topics-Edition/dp/0132306379
   
   You can find most of those in pdf format somewhere online. I'm not saying those are the best textbooks to use, but they should at least provide a guide so that you can be sure you taking the right courses.
   
   Edit: If your school offers a combined prob/stat course (usually offered for engineers) that has a calc 3 pre-req it would probably serve well as a compact introduction to the subject.
   

Spivak is probably the best intro to real analysis book out there, so I would say start there. Also, go dig out your old pre-calc textbook and be sure you've really mastered the basics. For diff eq, just keep your calc tools extra sharp - think deeply about the fundamental theorem and get right with taylor series. Maybe go back and work through the material on continuously compounding interest and radioactive decay, that sort of thing. For probability, start with khan and then try to find introduction to probability by Bertsekas at your library. Most importantly: have fun and try to solve something nifty with your new math-toys :-)

Lady Luck: The Theory of Probability, by Warren Weaver

http://www.amazon.com/Lady-Luck-Theory-Probability-Science/dp/0486243427

This is a great introduction to probability theory, an entertaining read, and written precisely for the motivated high school student. It also represents a nice little slice of history. Warren Weaver was an adviser of and co-author with Claude Shannon, the founder of modern information theory.

Here are some great books that I believe you may find helpful :)

• Elements of Algebra by Leonhard Euler
  
• Foundations of Mathematics by Ian Stewart
  
• Geometry Revisited by H. S. M. Coxeter
  
• Excursions in Calculus by Robert M. Young
  
• New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian
  
• The Calculus Gallery: Masterpieces From Newton to Lebesgue by William Dunham
  
• A Primer of Real Functions by Ralph P. Boas
  
• Understanding Analysis by Stephen Abbot
  
• Chaos: Making a New Science by James Gleick
  
• Logic: An Introduction to Elementary Logic by Wilfred Hodges
  
  and last but definitely not least:
  
  
• A Mathematical Gift I, II, III by Kenji Ueno et al.
  
  Later on:
  
  
• Conceptual Mathematics: A First Introduction to Categories by F. William Lawvere
  
• Twelve Landmarks of Twentieth-Century Analysis by D. Choimet & H. Queffélec
  
• Theory of Complex Functions by Reinhold Remmert
  
• Algebraic Combinatorics. Walks, Trees, Tableaux & More by Richard P. Stanley

> The same way that buying a lotto ticket this week doesn't increase my odds of winning next weeks draw if I don't win.

no, but buying multiple tickets this week increses your chance to win this week

> You would be correct if you could use multiple keys on one box to increase your odds of getting a ship, but thats not how it work.

yes, i am also correct if i can use multiple keys on multiple boxes

> You only get to buy 1 ticket to this weeks lotto, one for next weeks etc etc

not really, you can buy as many tickets as you can afford

i suggest https://www.amazon.com/Probability-Dummies-Deborah-J-Rumsey/dp/0471751413/ unless you want to make fool out of yourself some more...

http://www.amazon.com/Applied-Regression-Analysis-Generalized-Linear/dp/0761930426

http://www.amazon.com/Regression-Categorical-Dependent-Quantitative-Techniques/dp/0803973748

http://www-stat.stanford.edu/~tibs/ElemStatLearn/

http://www-bcf.usc.edu/~gareth/ISL/

http://www.amazon.com/Extending-Linear-Model-Generalized-Nonparametric/dp/158488424X/ref=sr_1_2?s=books&ie=UTF8&qid=1380716057&sr=1-2

http://www.amazon.com/Generalized-Edition-Monographs-Statistics-Probability/dp/0412317605

hope it helps man. good luck. im learning this stuff for my project too

I like to plug Euler's Gem when this formula arises. It's a great book with lots of wonderful applications.

> Probability states that either major party candidate had a 50% chance of winning (since there are two running,

Wow. Really? So all of probability theory is wrong then.

I recommend this book:
https://www.amazon.ca/Probability-Dummies-Deborah-J-Rumsey/dp/0471751413

Measure, Integral and Probability Pretty good book. As an undergrad I used that for the third in a sequence for analysis. The first was real in one variable with the first 9 chapters of Bartle, and the second was multi-dim with Spivak's little white book.

Introduction to Probability and Mathematical Statistics This is another senior level math book, but can be used as a first year intro grad level book. It seems people either love it or hate it. I really enjoyed it.

A first Course in Probability This is the standard book. Personally, I hate it. But, you can't love all the books!

Hope that helps.

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).

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

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).

For probability, go with Ross every time. Clear and concise, I've found no better book on probability and stochastic processes.

Intro to Probability:
http://www.amazon.com/First-Course-Probability-Sheldon-Ross/dp/013603313X/ref=sr_1_1?ie=UTF8&s=books&qid=1266746119&sr=8-1-spell

Another good introduction, with more advanced modeling topics:
http://www.amazon.com/Introduction-Probability-Models-Ninth-ebook/dp/B000WNFX3O/ref=sr_1_2?ie=UTF8&s=digital-text&qid=1266746119&sr=8-2-spell

Stochastic (random) Processes:
http://www.amazon.com/Stochastic-Processes-Sheldon-M-Ross/dp/0471120626/ref=sr_1_1?ie=UTF8&s=books&qid=1266746101&sr=8-1

Here is Occupational Health Psychology: https://www.amazon.com/Handbook-Occupational-Health-Psychology-Second/dp/1433807769/ref=sr_1_2?crid=23K4PM6UI8F10&keywords=handbook+of+occupational+health+psychology&qid=1574832541&sprefix=handbook+of+occupation%2Caps%2C198&sr=8-2

​

Here is also a great stats textbook: https://www.amazon.com/Discovering-Statistics-Using-IBM-SPSS/dp/1526436566/ref=sr_1_1?keywords=andy+field+statistics&qid=1574833320&sr=8-1

The same author also has a interesting version of a stats book: https://www.amazon.com/Adventure-Statistics-Reality-Enigma/dp/1446210456/ref=sr_1_3?keywords=andy+field+statistics&qid=1574833347&sr=8-3

For theoretical foundations and readability, the Theory of Probability: Explorations and Applications by Santosh S. Venkates is my favorite. Venkates' passion for the subject is contagious, and his historical references give a you additional perspective on the foundations of the subject https://www.amazon.com/Theory-Probability-Explorations-Applications/dp/1107024471/

For practical, no-nonsense approach with real-world examples John Bertsekas & Tsitsiklis' Intro to Probability is excellent (although in my opinion more of a manual and reference) https://www.amazon.com/Introduction-Probability-Dimitri-P-Bertsekas/dp/188652940X/

I mean it in a kind of casual sense. It seems like after reading and writng so much mathematics, a lot of people converge to a writing style that's characteristic to math writing. They'll default to writing in first person plural whenever possible ("we have that...", "we define"), they'll use the word "therefore" almost every other sentence, and they'll generally write in a rather dry but logically clear manner.

I'd contrast this with the more "didactic" writing style of the usual high school texts (e.g. something like this.)

I'm a math tutor and I use these books with almost all my students. They go into pretty good detail about the why's and have quite challenging problems. They include chapter tests, chapter reviews, tons of word problems, challenging multiple SAT-type questions every other chapter, and cumulative reviews. They also thoroughly prepare you for every calculus topic, as well as probability and statistics. They cover Algebra through Pre-calc:

Algebra

Geometry

Algebra 2

Precalculus

To supplement those, you could also use this British math series. It should fill in any possible gaps or clarify certain topics:

9th grade

10th/11th grade

12th grade

By far, the best resource is Andrew Gelman's book Data Analysis Using Regression and Multilevel/Hierarchical Models.

Also, the book Extending the Linear Model with R by Julian Faraway touches on it.

Other than that, here are some blog posts that I found helpful:

https://ourcodingclub.github.io/2017/03/15/mixed-models.html

https://www.jaredknowles.com/journal/2013/11/25/getting-started-with-mixed-effect-models-in-r

http://www.bodowinter.com/tutorial/bw_LME_tutorial2.pdf

And here's the paper written by Doug Bates, the author of the r mixed effects package lme4.

http://www.amazon.com/Introduction-Probability-Dimitri-P-Bertsekas/dp/188652940X

Chapter 4 and beyond should be a good point to start from where you leave off. There's important stuff like convergence and then stochastic processes. You shouldn't need an analysis or measure theory class to understand the material but it would be helpful.

I'd recommend Conway and Guy's The Book of Numbers and Abbott's Flatland: A Romance of Many Dimensions.

The Book of Numbers by John Conway. Its not particularly "real-world" but it is an excellent book.

Can you help me find these books?

https://www.amazon.com/Stats-Models-Richard-D-Veaux/dp/0321986490 Stats: Data and Models (4th Edition)

ISBN: 9780321986498

Author: Richard D. De Veaux

https://www.amazon.com/Proud-Shoes-Black-Women-Writers/dp/0807072095
Proud Shoes : The Story Of An American Family

ISBN: 9780807072097

Author: Murray

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.

Ross for probability. He wrote the undergrad book on probability. It is on it's 9th edition, so you can probably find an older edition for next to nothing.

Statistics for engineers and scientists by Devore would probably be a good book for learning stats as a physicist. I've taught out of it a few times and I like it as book. Again, you can find an international edition of this book for next to nothing.

Casella and Berger is a first year grad text/ upper level undergrad text. You need some mathematical maturity to do use it and it probably goes a lot deeper than you would want. I would not suggest it.

The best introduction to probability is A First Course in Probability by Ross, in my opinion.

thanks for that info. the only reason i wanted to brush up on calc was to maybe then go and study statistics from a more advanced book. those usually have proofs that use calc. i was going to study from this book: https://www.amazon.com/Stats-Models-Richard-D-Veaux/dp/0321986490/ref=sr_1_1?ie=UTF8&qid=1491175118&sr=8-1&keywords=9780321986498

It does not, and the set of confounding variables in fitness is legion. The particular study you linked is a pure observational exercise that makes no attempt to deal with selection issues and the idea that one might find causality in the results is fantasy.

Consider checking out some causality literature, either the Ruben-Imbens thread or the Pearl devotees. Join us in believing nothing very few extant studies.

Or just read people like Andrew Gelman and dispense with learning about causality, there are ten dozen other reasons most published research findings are false.

Related.

Here is a free resource I have used from time to time.
And another.

I have also been recommended this textbook by lecturers, never actually picked it up though.

They are probably a bit advanced but should at least help you understand something about those fancy numbers.

https://www.amazon.com/Probability-Dummies-Deborah-J-Rumsey/dp/0471751413
https://www.amazon.com/Statistics-Dummies-Lifestyle-Deborah-Rumsey/dp/1119293529

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.

I read Sternberg as an undergrad and loved it. It's very accessible.

I found this book to be quite a bit of fun after I finished calculus. http://www.amazon.com/Excursions-Calculus-Continuous-Mathematical-Expositions/dp/0883853175
Good expository with lots of fun problems that only require calculus, but covers all sorts of topics.

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 Trump supporters say this, they might as well wear a neon sign that says "I lack a 5th grade education in probability and statistics!"

Read this, if you can:

https://www.amazon.com/Probability-Dummies-Deborah-J-Rumsey/dp/0471751413

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.

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

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

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.

For research methods in behavioral and social sciences, you probably can't get better than Shadish, Cook, and Campbell's : Experimental and Quasi-Experimental Design for generalized causal inference. As far as stats go, the Andy Field books are good and he has one for R, one for SAS, and one for SPSS. I prefer the John Fox book on Applied Regression analysis and the corresponding r book. Here are some links:
http://www.amazon.com/Experimental-Quasi-Experimental-Designs-Generalized-Inference/dp/0395615569/ref=sr_1_1?ie=UTF8&qid=1417220301&sr=8-1&keywords=shadish+cook+and+campbell

http://www.amazon.com/Applied-Regression-Analysis-Generalized-Linear/dp/0761930426/ref=sr_1_6?ie=UTF8&qid=1417220395&sr=8-6&keywords=John+Fox

http://www.amazon.com/An-R-Companion-Applied-Regression/dp/141297514X/ref=pd_bxgy_b_img_y

Hey, a probability and geometry book is on sale as well.

Then there's "Mathematics for Non-mathematicians"

I like Faraway as an introduction, it's very approachable.

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.

Book is An Adventure in Statistics: The Reality Enigma: https://www.amazon.com/Adventure-Statistics-Reality-Enigma/dp/1446210456

Euler's Gem by David Richeson.

This follows the story of how some deep insights of Leonard Euler led to the creation of the mathematical field of topology. Topology is the mathematics of space when we think only about the connections between points, rather than their precise distance (as in geometry). A great example is the London tube map. Many mathematical questions that are impossible in geometry can be considered in this way.

I would recommend this book especially for those in school who suspect there is more to maths than crunching numbers, but also to those who have never enjoyed the subject but are willing to see it in a new light.

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.

ISBN: 9780395977255
Title: Algebra and Trigonometry, Grades 10-12 Structure and Method Book 2: Mcdougal Littell Structure & Method
Author: Houghton Mifflin Company
Copyright: 1/26/99
Publisher: HBC
Amazon link: https://www.amazon.com/dp/0395977258/

CLRS for algorithms/CS.

Probability and random processes for statistics.

Biological Sequence Analysis by Richard Durbin for my subfield of bioinformatics.

That's the name of the book. Yet Another Introduction to Analysis

Pm me i'll order this for you

https://www.amazon.com/Probability-Dummies-Deborah-J-Rumsey/dp/0471751413

A Primer of Real Functions - Boas

Uhm. Do you mean this one?

https://www.amazon.com/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220

Try this.

Imbens and Rubin have a relatively new book:

Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction

http://www.amazon.com/Causal-Inference-Statistics-Biomedical-Sciences/dp/0521885884

Conway and Guy's The Book of Numbers

for you because you seem upset

Perhaps this or this.

Grimmett and Stirzaker.

http://www.amazon.com/Lady-Luck-Theory-Probability-Mathematics/dp/0486243427 This is the one.

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.

Dude, Feller

http://www.amazon.com/Introduction-Probability-Theory-Applications-Vol/dp/0471257087

This one?

https://amzn.com/052138835X

/u/istvan_magyary have you tried libgen?

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.

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

No the drop isn't broken. Here, this should help.

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...