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I never had the opportunity to visit a stats course from a math faculty. I am looking for a probability theory and statistics book that is complete and self-sufficient. By complete I mean that it contains all the proofs and not just states results. By self-sufficient I mean that I am not required to read another book to be able to understand the book. Of course it can require college level (math student) calculus and linear algebra.

I have looked at multiple books and I didn't like any of them.

"Weighing the Odds" from David Williams is more formal than DeGroot and seems to be complete and self-sufficient. However, I find the style strange. He also invents new terms that only he seems to use. All the stuff that is explained in DeGroot too is explained better there.

If you know a great book in German that's also fine as I am German.

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    $\begingroup$ What level of text are you looking for? I think that Degroot book is aimed more at undergraduate students. A good book for graduate level studies is Statistical Infernece by Casella and Berger. $\endgroup$
    – user25658
    Sep 19, 2013 at 22:17
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    $\begingroup$ This definition of "self sufficient" is subjective, because your ability to "understand the book" depends on your background. $\endgroup$
    – whuber ♦
    Sep 20, 2013 at 1:44
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    $\begingroup$ I'm guessing that there is no book that you will find completely satisfactory. $\endgroup$
    – mark999
    Sep 20, 2013 at 9:18
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    $\begingroup$ Self sufficient given the knowledge that you have after obtaining a bachelor in mathematics. With regards to the topics Degroot is what I am looking for but I don't like books in which core results (e.g. chi square distribution of the test statistics given the null hypothesis is true for the likelihood ratio test) are not derived. I will have a look at Statistical Inference by Casella and Berger. $\endgroup$
    – Julian Karch
    Sep 20, 2013 at 11:25
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    $\begingroup$ How can a book on probability and statistics ever be complete? Even huge multi-volume tomes (Kendall and Stuart's .. etc's Advanced theory of Statistics in its latest incarnations, for example, come to thousands of pages if I recall correctly) aren't remotely complete. $\endgroup$
    – Glen_b
    Sep 21, 2013 at 1:36

8 Answers 8

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If you are searching for proofs, I have been working for some time on a free stats textbook that collects lots of proofs of elementary and less elementary facts that are difficult to find in probability and statistics books (because they are scattered here and there). You can have a look at it at http://www.statlect.com/

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  • $\begingroup$ Why are they scattered here and there? I thought teaching statistics is supposed to be systematic. $\endgroup$
    – Cheng
    Apr 27 at 0:17
  • $\begingroup$ @Cheng. It probably is, statistically speaking. $\endgroup$
    – Mad Physicist
    Jun 12 at 2:12
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If you want to read probability as a story, read the best book ever by Feller. I am also guessing that you do not want to go to the level of measure theoretic definition of probabilities which has specialized books. another beginner level book is from Ross. Other specialized applications have specialized books. so more information will gather better suggestions.

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I would recommend two books not mentioned, as well as several already mentioned.

The first is E.T. Jaynes "Probability: The Language of Science." It is polemic and he is a very partisan author, but it is very good.

The second is Leonard Jimmie Savage's "The Foundations of Statistics." You will probably be very surprised when you first start reading it as you will not expect it to go the route it goes.

Both are writing foundational work in Bayesian probability and Bayesian statistics. The above works are non-Bayesian.

Both books are completely contained and self-sufficient. Indeed, they build from the foundation upward. Both approach it axiomatically.

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    $\begingroup$ Well don't leave us in suspense, what is the unexpected route that Savage's book follows? $\endgroup$
    – Praxeolitic
    Oct 26, 2017 at 1:40
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    $\begingroup$ @Praxeolitic Savage grounds his book in preference theory. You construct a strictly "personalistic" basis for probability and statistics. What is as interesting is that these measures are intrinsically admissible statistics, whereas that is not automatically true for non-Bayesian methods. $\endgroup$
    – Dave Harris
    Nov 8, 2017 at 0:04
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Finding a single, comprehensive book will be very difficult. If you're asking because you want to do some self-study, get a couple of used texts instead of a single new one. You can get classics for $3-10 dollars if you look around online.

Feller's "Introduction to Probability" is great for its completeness and expository style, but I don't like the exercises much. And the exposition would not make it so good for a reference. He tends to have a lot of long examples, which is great for fostering understanding, and not so great for looking things up.

I enjoyed Allan Gut's "An Intermediate Course in Probability". There is some overlap with Feller, but it goes into greater depth on those topics. He covers the various transformations, order statistics (which, if I recall, Feller only does by example).

Ross' Introduction to Probability Models is pretty comprehensive, but it is very example oriented. Again, that is not my favorite style (I'd rather they saved those examples for exercises with hints, and kept them out of the main flow), but if it works for you, I can recommend it.

You might as well consider Cacoullos' "Exercises in Probability" and Mosteller's "50 Challenging Exercises in Probability".

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For the probability side I like Probability and Random Processes by Grimmett & Stirzaker. It has a nice way of giving intuitive explanations whilst still being fairly rigorous and providing some proofs at least.

For the Statistics side I've had Theory of Statistics by Schervish on my wish list for a while now but not got around to buying it, so I can only say I've heard good things about it...it's supposed to be a graduate level introduction so possibly more rigorous than the other Schervish book you mention.

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  • $\begingroup$ +1 for Theory of Statistics by Schervish. It is an excellent book for anyone who is well versed with measure-theoretic probability and wants an almost complete statistics book. $\endgroup$
    – Aditya
    Sep 24, 2020 at 3:09
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I recommend Probability Theory and Mathematical Statistics by Marek Fisz, because:

  1. It contains most of the common proof, but without making the book too difficult as an introduction book
  2. It is quite theoretical, but still contain enough well-designed examples to illustrate points
  3. Exercises are meaningful. Some of them are more advanced famous results
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As noted by many others, there is no single good text for any scientific subject simply because any given authors or group of authors use a set of assumptions regarding the readers' level of understanding and diversity of knowns and unknowns in the user's brain. Said this, my suggestion for someone knows basics in calculus and linear algebra is to begin with the "modern mathematical statistics with applications" by Devore and Berk.

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  • $\begingroup$ Since you mentioned Devore, I want to know your opinion of his other well-known book : Probability and Statistics for Engineering and the Sciences by Jay L. Devore. Among other similar books, I think I will use it as the main textbook. Others such as by Sheldon Ross's book, Morris DeGroot's book, Miller and Freunds's book are really interesting, though, but I will go to the former first. I had learnt this subject about 33 years ago, but mostly for passing the exam. So I need to relearn it in the correct manner. $\endgroup$
    – Lex Soft
    Jul 6, 2022 at 2:56
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You can read Student's Solutions Guide for Introduction to Probability, Statistics, and Random Processes book. It provides clear examples and exercises with "additional questions" at the end of each chapter which really help improve learning and there is a logical progression from one idea to another.

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