Probability Theory: A Concise Course - Dover Mathematics Book | Perfect for Students & Self-Learners | Study Guide for Statistics & Data Science

"Concise" is indeed the operative word here. This book is probably not suitable as a first text on the subject, but makes an excellent review or quick reference for the topics it covers.Essentially, this text is geared toward taking someone who has - in principle - no knowledge of probability and introducing them specifically to Markov processes. There is very little attention paid to conditional probabilities, and Bayes' rule is never even mentioned.Also, this book requires no measure theory.Chapter 1 covers basic concepts: probability as relative frequency, sampling with and without replacement, binomial and multinomial coefficients.Chapter 2 is titled "Combination of Events". It introduces the idea of the sample space, and focuses on how probability interacts with set theoretic operations such as intersection and union. It ends with a proof of the First Borel-Cantelli Lemma.The third chapter introduces independence and ends with a proof of the Second Borel-Cantelli Lemma.The Borel-Cantelli Lemmas are somewhat technical results that are needed to the get the theory of Markov processes off the ground, so it's pretty clear where this book is headed early on. The proofs of both of the lemmas are very tidy.Chapter 4 is devoted to random variables. Here we find the definitions of expectation, variance, and the correlation coefficient along with Chebyshev's Inequality.Chapter 5 covers the Bernoulli distribution, the Poisson distribution, and the Normal distribution. We are also treated to the De Moivre-Laplace theorem as a stepping stone toward the Central Limit Theorem.Chapter 6 is titled "Some Limit Theorems". We are immediately provided with the proof and then statement - in that order - of the Weak Law of Large Numbers. We are then provided merely with the statement of the Strong Law of Large Numbers. This chapter then introduces Generating Functions which are used quite heavily in the remainder of the work. This chapter also introduces Characteristic Functions, which don't get much attention and concludes with the Central Limit Theorem.Chapter 7 introduces Markov Chains while chapter 8 covers Continuous Markov Processes and naturally covers the Chapman-Kolmogorov equations. Here simply called the Kolmogorov equations for the fairly obvious reason that the author is Russian.The book ends with four short appendices which introduce the reader in turn to the following topics: Information Theory, Game Theory, Branching Processes, and Optimal Control. I thought these were wonderful although obviously none of them covers very much ground.This book is actually quite delightful especially for someone who already has some background in basic probability. It does provide and good and very quick introduction to Markov processes, but it's scope of coverage of any topic is necessarily quite limited.