Recommendation for probabilities books

Question

I am a university student, majoring in Mathematics, I have previously studied "a text book of convergence" by W. L. Ferrare, which is the best book I have studied , because it has many advantages such as simple English language and suitable for all readers, and others advantages in how present the ideas.... ., Now I need a book like it, But in another branch, which is the probabilities , So I want some of your suggestions

Answer 1

The central characteristics of Ferrar's book are that it's a concise introduction to the modern, rigorous viewpoint on convergence of sequences and series, including power series.

The problem with probability theory is that the correct rigorous approach to the subject is at a considerably higher level of sophistication, as it involves measure theory. But if this isn't a problem, a similar approach would be the concise "Probability with Martingales" by David Williams.

This book is limited as it doesn't have many of the typical beginner's examples in probability theory. For this, good parallel reading would be "Weighing the Odds," by the same author. In fact, this lower-level book often refers to "Probability with Martingales" for proofs on more theoretical points.

A different approach is to remain rigorous but avoid measure theory altogether by focusing only on countable probability spaces, excluding the continuous theory. This is done in Volume 1 of Feller's "Introduction to Probability Theory and its Applications", which is known for its difficult exercises. The author's goal is to illustrate many of the more important non-trivial phenomena in probability theory while remaining within the realm of discrete sample spaces. In a way, this would be a good follow-on from Ferrar because you'll already be familiar with all the theorems you need on convergence of series.

Answer 2

Probability and Statistics for Engineers and Scientists by Ronald E. Walpole, Raymond Myers and Sharon L. Myers is good.

Answer 3

A Second Course in Probability by Ross is a good place to start, if you're interested in a book which explains the fundamental concepts well in simple English, but some of the proofs are either explained heuristically, or referred to some other textbook.

Another good book which has served me well is Probability: Theory and Examples by Durrett. Most graduate courses on measure theoretic probability use it, and the explainations are quite good in my opinion, although they tend to leave some of the more trivial proofs as exercises. I believe that this book is a good starting point for anyone interested in probability research.

Probability: A Graduate Course follows roughly the same flow as Durrett, but it makes a point of proving almost every result, even the more trivial ones although it covers fewer topics (iirc the usual Law of Large Numbers, Central Limit Theorem, Martingales). With some background in measure theory, this is a good book for self study. I personally used it during my undergraduate studies, when self studying several topics for my final year thesis. Regretfully I did not take any courses in measure theory back then, but this book was still quite accessible after going through the chapter on introductory measure theory.