A First Course in Probability and Markov Chains

Preface

This book collects topics covered in introductory courses in probability delivered by the authors at the University of Florence. It aims to introduce the reader to typical structures of probability with a language appropriate for further advanced reading. The attention is mainly focused on basic structures.

There is a well established tradition of studies in probability due to the wide range of possible applications of related concepts and structures in science and technology. Therefore, an enormous amount of literature on the subject is available, including treatises, lecture notes, reports, journal papers and web pages. The list of references at the end of this book is obviously incomplete and includes only references used directly in writing the following pages. Throughout this book we adopt the language of measure theory (relevant notions are recalled in the appendices).

The first part of the book deals with basic notions of combinatorics and probability calculus: counting problems and uniform probability, probability measures, probability distributions, conditional probability, inclusion–exclusion principle, random variables, dispersion indexes, independence, and the law of large numbers are also discussed. Central limit theorem is presented without proof. Only a basic knowledge of linear algebra and mathematical analysis is required.

In the second part we discuss, as a first example of stochastic processes, Markov chains with discrete time and discrete states, including ...