Introduction to Probability: Models and Applications

Book Preface

Probability theory deals with phenomena whose outcome is affected by random events, and therefore they cannot be predicted with certainty. For example, the result of throwing a coin or a dice, the time of occurrence of a natural phenomenon or disaster (e.g. snowfall, earthquake, tsunami etc.) are some of the cases where “randomness” plays an important role and the use of probability theory is almost inevitable.

It is more than five centuries ago, when the Italians Luca Pacioli, Niccolo Tartaglia, Galileo Galilei and the French Pierre de Fermat and Blaise Pascal started setting the foundations of probability theory. Nowadays this area has been fully developed as an independent research area and offers valuable tools for almost all applied sciences. As a consequence, introductory concepts of Probability Theory are taught in the first years of most University and College programs.

This book is an introductory textbook in probability and can be used by majors in Mathematics, Statistics, Physics, Computer Science, Actuarial Science, Operations Research, Engineering etc. No prior knowledge of probability theory is required. In most Universities and Colleges where an introductory Probability course, such as one that may be based on this textbook, is offered, it would normally follow a rigorous Calculus course. Consequently, the Probability course can make use of differential and integral calculus, and formal proofs for theorems and propositions may be presented to the students, thereof offering them a mathematically sound understanding of the field. For this reason, we have taken a calculus-based approach in this textbook for teaching an introductory course on Probability. In doing so, we have also introduced some novelties hoping that these will be of benefit to both students and instructors.

In each chapter, we have included a section with a series of examples/problems for which the use of a computer is required. We demonstrate, through ample examples, how one can make effective use of computers for understanding probability concepts and carrying out various probability calculations. For these examples it is suggested to use a computer algebra software such as Mathematica, Maple, Derive, etc. Such programs provide excellent tools for creating graphs in an easy way as well as for performing mathematical operations such as derivation, summation, integration, etc; most importantly, one can handle symbols and variables without having to replace them with specific numerical values. In order to facilitate the reader, an example set of Mathematica commands is given each time (analogous commands can be assembled for the other programs mentioned above). These commands may be used to perform a specific task and then various similar tasks are requested in the form of exercises. No effort is made to present the most effective Mathematica program for tackling the suggested problem and no detailed description of the Mathematica syntax is provided; the interested reader is referred to the Mathematica Instruction Manual (Wolfram Research) to check the, virtually unlimited, commands available in this software (or alternative computer algebra software) and use them for creating several alternative instruction sets for the suggested exercises.

Moreover, a novel feature of the book is that, at the end of each chapter, we have included a section detailing a case study through which we demonstrate the usefulness of the results and concepts discussed in that chapter for a real-life problem; we also carry out the required computations through the use of Mathematica.

At the beginning of each chapter we provide a brief historical account of some pioneers in Probability who made exemplary contributions to the topic of discussion within that chapter. This is done so as to provide students with a sense of history and appreciation of the vital contributions made by some renowned probabilists. Apart from the books on the history of probability and statistics that can be found in the bibliography, we have used Wikipedia as a source for biographical details.

In most sections, the exercises have been classified into two groups, A and B. Group A exercises are usually routine extensions of the theory or involve simple calculations based on theoretical tools developed in the section and should be the vehicle for a self-control of the knowledge gained so far by the reader. Group B exercises are more advanced, require substantial critical thinking and quite often include fascinating applications of the corresponding theory.

In addition to regular exercises within each chapter, we have also provided a long list of True/False questions and another list of multiple choice questions. In our opinion, these will not only be useful for students to practice with (and assess their progress), but can also be helpful for instructors to give regular in-class quizzes.

Particular effort has been made to give the theoretical results in their simplest form, so that they can be understood easily by the reader. In an effort to offer the book user an additional means of understanding the concepts presented, intuitive approaches and illustrative graphical representations/figures are provided in several places. The material of this book emerged from a similar book (Introduction to Probability: Theory and Applications, Stamoulis Publications) written by one of us (MVK) in Greek, which is being used as a textbook for many years in several Greek Universities. Of course, we have expanded and transformed this material to reach an international audience.

This is the first volume in a set of two for teaching probability theory. In this volume, we have detailed the basic rules and concepts of probability, combinatorial methods for probabilistic computations, discrete random variables, continuous random variables, and well-known discrete and continuous distributions. These form the core topics for an introduction to probability. More advanced topics such as joint distributions, measures of dependence, multivariate random variables, well-known multivariate discrete and continuous distributions, generating functions, Laws of Large Numbers and the Central Limit Theorem should come out as core topics for a second course on probability. The second volume of our set will expand on all these advanced topics and hence it can be used effectively as a textbook for a second course on probability; the form and structure of each chapter will be similar to those in the present volume.

We wish to thank our colleagues G. Psarrakos and V. Dermitzakis who read parts of the book and to our students who attended our classes and made several insightful remarks and suggestions through the years.

In a book of this size and content, it is inevitable that there are some typographical errors and mistakes (that have clearly escaped several pairs of eyes). If you do notice any of them, please inform us about them so that we can do suitable corrections in future editions of this book.

It is our sincere hope that instructors find this textbook to be easy-to-use for teaching an introductory course on probability, while the students find the book to be user-friendly with easy and logical explanations, plethora of examples, and numerous exercises (including computational ones) that they could practice with!

Finally, we would like to thank the Wiley production team for their help and patience during the preparation of this book!
March, 2019 N. Balakrishnan
Markos V. Koutras
Konstadinos G. Politis