A Natural Introduction to Probability Theory

In this introduction to probability theory, we deviate from the route usually taken. We do not take the axioms of probability as our starting point, but re-discover these along the way. First, we discuss discrete probability, with only probability mass functions on countable spaces at our disposal. Within this framework, we can already discuss random walk, weak laws of large numbers and a first central limit theorem. After that, we extensively treat continuous probability, in full rigour, using only first year calculus. Then we discuss infinitely many repetitions, including strong laws of large numbers and branching processes. After that, we introduce weak convergence and prove the central limit theorem. Finally we motivate why a further study would require measure theory, this being the perfect motivation to study measure theory. The theory is illustrated with many original and surprising examples.

Table of Contents:
Zentralblatt Math"Most textbooks designed for a one-year course in mathematical statistics cover probability in the first few chapters as preparation for the statistics to come. This book in some ways resembles the first part of such textbooks: it's all probability, no statistics. But it does the probability more fully than usual, spending lots of time on motivation, explanation, and rigorous development of the mathematics?. The exposition is usually clear and eloquent?. Overall, this is a five-star book on probability that could be used as a textbook or as a supplement." MAA online

Review :
"The book [is] an excellent new introductory text on probability. The classical way of teaching probability is based on measure theory. In this book discrete and continuous probability are studied with mathematical precision, within the realm of Riemann integration and not using notions from measure theorya ]. Numerous topics are discussed, such as: random walks, weak laws of large numbers, infinitely many repetitions, strong laws of large numbers, branching processes, weak convergence and [the] central limit theorem. The theory is illustrated with many original and surprising examples and problems." a "ZENTRALBLATT MATH "Most textbooks designed for a one-year course in mathematical statistics cover probability in the first few chapters as preparation for the statistics to come. This book in some ways resembles the first part of such textbooks: it's all probability, no statistics. But it does the probability more fully than usual, spending lots of time on motivation, explanation, and rigorous development of the mathematicsa ]. The exposition is usually clear and eloquenta ]. Overall, this is a five-star book on probability that could be used as a textbook or as a supplement." a "MAA ONLINE "It seems that a task to provide an introductory course on probablitity fulfilling the following requirements arises not so rarely: (A) The course should be accessible to studnets having only very modest preliminary knowledge of calculus, in particular, with no acquaintance with measure theory. (B) The presentation should be fully rigorous. (C) Nontrivial resuilts should be give. (D) Motivation for further strudy of measure theoretic probability ought to be provided, hence to contetnoneself to countable probability spaces is undesirable. R. Meester's book is an attametp to shot that all these demands may be fulfilled in a reasonalb eway, however incompatible they may look at first sight." ---Mathematica Bohemica "The book [is] an excellent new introductory text on probability. The classical way of teaching probability is based on measure theory. In this book discrete and continuous probability are studied with mathematical precision, within the realm of Riemann integration and not using notions from measure theory???. Numerous topics are discussed, such as: random walks, weak laws of large numbers, infinitely many repetitions, strong laws of large numbers, branching processes, weak convergence and [the] central limit theorem. The theory is illustrated with many original and surprising examples and problems." ???ZENTRALBLATT MATH "Most textbooks designed for a one-year course in mathematical statistics cover probability in the first few chapters as preparation for the statistics to come. This book in some ways resembles the first part of such textbooks: it's all probability, no statistics. But it does the probability more fully than usual, spending lots of time on motivation, explanation, and rigorous development of the mathematics???. The exposition is usually clear and eloquent???. Overall, this is a five-star book on probability that could be used as a textbook or as a supplement." ???MAA ONLINE "It seems that a task to provide an introductory course on probablitity fulfilling the following requirements arises not so rarely: (A) The course should be accessible to studnets having only very modest preliminary knowledge of calculus, in particular, with no acquaintance with measure theory. (B) The presentation should be fully rigorous. (C) Nontrivialresuilts should be give. (D) Motivation for further strudy of measure theoretic probability ought to be provided, hence to contetn oneself to countable probability spaces is undesirable. R. Meester's book is an attametp to shot that all these demands may be fulfilled in a reasonalb eway, however incompatible they may look at first sight." ---Mathematica Bohemica "The book Ýis¨ an excellent new introductory text on probability. The classical way of teaching probability is based on measure theory. In this book discrete and continuous probability are studied with mathematical precision, within the realm of Riemann integration and not using notions from measure theory. Numerous topics are discussed, such as: random walks, weak laws of large numbers, infinitely many repetitions, strong laws of large numbers, branching processes, weak convergence and Ýthe¨ central limit theorem. The theory is illustrated with many original and surprising examples and problems." ZENTRALBLATT MATH "Most textbooks designed for a one-year course in mathematical statistics cover probability in the first few chapters as preparation for the statistics to come. This book in some ways resembles the first part of such textbooks: it's all probability, no statistics. But it does the probability more fully than usual, spending lots of time on motivation, explanation, and rigorous development of the mathematics. The exposition is usually clear and eloquent. Overall, this is a five-star book on probability that could be used as a textbook or as a supplement." MAA ONLINE