I am trying to build a foundation for data science through self study. Is there a book that could teach me the fundamentals of statistics along with the related mathematical concepts in a comprehensive manner? Preferably one that has sufficiently detailed solved examples.
I'm assuming you are asking for beginner-level texts. I used a combination of books when I was an undergraduate in the Statistics Department. These are, in order of personal preference:
- Casella & Berger's Statistical Inference.
- Carol Ash's The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) (After struggling in probability as a Freshman, this book finally made me "get it". I wish more books were written like this today. Many, many worked examples thoroughly explained. RIP Carol Ash).
- Marco Taboga's Lectures on Probability Theory and Mathematical Statistics (great for proofs of important statistical results).
- Rohatgi and Ehsanes Saleh's An Introduction to Probability and Statistics, 3rd. (More advanced)
- Leadbetter, Cambanis, and Pipiras' A Basic Course in Measure and Probability: Theory for Applications. (a must read course in measure theory)
- Nitis Mukhopadhyay's Probability and Statistical Inference. (an excellent underrated text in my opinion).
- Johnson and Bhattacharyya's Statistics: Principles and Methods, 3rd. (very elementary/basic but great beginning book if no previous exposure).
- George Roussas' An Introduction to Probability and Statistical Inference. 2ed.
- Richard A. Johnson and Dean W. Wichern's Applied Multivariate Statistical Analysis, 6th ed. (more specialized in multivariate methods)
- John E. Freund's Mathematical Statistics, 8th ed. (good example problems).
- Wackerly, Mendenhall, and Scheaffer's Mathematical Statistics with Applications, 7th. (good for basic example problems)
- Larry Wasserman's All of Statistics: A Concise Course in Statistical Inference.
- Ramachandran and Tsokos' Mathematical Statistics with Applications.
- Bain and Engelhardt's Introduction to Probability and Mathematical Statistics (lots of basic worked examples).
- Richard McElreath's Statistical Rethinking: A Bayesian Course with Examples in R and Stan. (Modern Bayesian Statistics Course with examples in R, obviously).
Also, I'd highly recommend you gain a basic understanding of linear algebra and multivariate calculus. For these, I recommend:
I have not included any books on statistical subspecialties such as regression, survey sampling, longitudinal data, etc. These are basic intro to mathematical statistics books. Keep in mind, when I majored in Statistics as an undergrad, that was two decades ago, and so computer programs were not found in most textbooks (or they were in SAS, sometimes in Fortran, and infrequently in S+). Many of the books listed above likely have newer editions that contain program code or companions websites now that probably provide more in the way of modern computer code for results. That being said, I think it's imperative to have a fundamental understanding of the statistical computations without a computer before progressing onto modern computational statistics.