PREFACE

This book is a follow-up of the book Introduction to Probability: Models and Applications, written by the same authors (which is referred throughout as Volume I). It thus forms the second volume for teaching probability theory.

In the first volume, we discussed the basic rules and concepts of probability, introduced the notion of conditional probability and independence, presented several combinatorial methods for probabilistic computations and finally went on to the introduction and study of discrete and continuous random variables; besides the presentation of general properties of random variables, we also discussed some well-known discrete and continuous distributions. These topics, in our view, form the core for an introduction to probability. The present volume discusses more advanced topics such as joint distributions, measures of dependence, multivariate random variables, some well-known multivariate discrete and continuous distributions, generating functions, laws of large numbers and the central limit theorem, which are the key topics for a second course on probability.

The form and structure of each chapter are similar to those in Volume I. In the beginning of each chapter, we provide a brief historical account of some pioneers in Probability who made exemplary contributions to the topic discussed in that chapter. This is done in order to provide students with a feeling of the history of Probability Theory and an appreciation of the vital contributions made by some renowned probabilists. Of course, books on the history of Probability and Statistics, would provide more elaborate details on their lives and contributions!

At the end of each section, the exercises have been classified into two groups, Group A and Group B. Exercises in Group A are usually routine extensions of the theory, involve simple calculations based on the theoretical tools presented in that section and are meant to consolidate the knowledge gained by the reader. Exercises in Group B are more advanced and would require both critical thinking and an ability to understand and use in an appropriate way the corresponding theory. In addition to regular exercises, we have also provided in each chapter, a long set of True/False questions and a set of multiple-choice questions. In our view, these will not only be useful for students to practice with (and assess their progress) but also for instructors to conduct regular in-class quizzes.

Special effort has been made to give the theoretical results in their simplest forms, so that they can be understood easily by the reader. In an effort to offer additional means for understanding the concepts presented, intuitive approaches as well as illustrative graphical representations/figures have been used throughout.

In each chapter, we have also included a section with some examples/problems for which the use of a computer is necessary. 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, the use of computer algebra software, such as Mathematica, Maple, and Derive, is recommended. These programs provide excellent tools for creating graphs in an easy way as well as for performing mathematical operations such as derivative, summation, integration, etc.; most importantly, one can handle symbols and variables without having to replace them with specific numerical values. To assist in this process, an example set of Mathematica commands is given each time (analogous commands can be assembled for other programs mentioned above as well). These commands may be used to perform a specific task and then several other similar tasks are set forth 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 may refer to the Mathematica Instruction Manual (Wolfram Research) to check the, virtually unlimited, commands available in this software package (or any other computer algebra software) and use them for creating several alternative instruction sets for the suggested exercises.

Moreover, as in the first volume, at the end of each chapter, we have included a section detailing a case study (application) through which we demonstrate the usefulness of the results and concepts discussed in that chapter for a real-life problem.

In the first volume, we focused on situations wherein we assigned probabilities to events with regard to a single random variable X. In particular, we distinguished between discrete and continuous random variables and discussed the most important probability distributions in both cases. Even though there were examples in which more than one variable can be defined within the same random experiment, we were only interested in how each of these variables varied individually. However, there are many instances wherein our interest may be to study the way two or more variables vary simultaneously and, as a result, our primary objective will be to formulate, and answer, probability statements concerning more than one variable. For example, suppose a company wishes to forecast its annual turnover, along with its total expenses, for next year. Let X denote the company's turnover and Y the company's expenses; then, our interest may be on probability statements of the form

for some intervals (A and B) on the positive half-line. If the company sets as a target the turnover to be at least x and the expenses to be at most y (x and y are specific values), then the probability that both these targets of the company will be met is

When x and y are treated as fixed, problems of this type have already been considered in Volume I. To be specific, let A be the event {X?=?x} and B be the event {Y?=?y}. Then, we simply want to find the probability that A and B occur at the same time, which is simply P(A ?n? B). However, and in analogy with the case of a single variable (i.e. univariate case), the company for its financial planning may want to see how the probability in the above equation varies for different values of x and y. This is accomplished with the concept of joint distribution of X and Y, which would explain how the two variables vary together. The central role played by probability distributions in the first volume is now extended to joint probability distributions of two or more variables, referred to as multivariate probability distributions. The particular case of two variables (known as the bivariate case) deserves special consideration and is therefore treated separately. This makes the transition from the univariate to the multivariate case smoother, and helps one to get a better understanding about the main changes needed to pass from the case of one variable to the multivariate case. For this reason, we have adhered to this structure, and in the first two chapters, we discuss (discrete and continuous) bivariate random variables and distributions, while in the third chapter, we treat the multivariate case, along with the fundamental concept of independence between random variables. In Chapter 4, we present techniques for obtaining the distribution of transformations of random variables, and then we use them to derive three additional distributions that are of special importance in Statistics.

The presentation of the concept of independence between random variables in Chapter 3, is restricted to the consideration whether two or more variable are independent or not. However, if two random variables are not independent, we would naturally be interested to introduce a measure to quantify their relationship. In Chapter 5, we discuss two such measures, called covariance and correlation. In Chapter 6, we present in detail some of the most important multivariate distributions. The last two chapters of the book deal with two areas of probability theory that provide essential tools for handling a wide range of applications: generating functions (Chapter 7) and limit theorems (Chapter 8). Their common appeal is in situations when one is interested in sums or averages of random variables. For example, when we want to estimate the mean from a (potentially large) population, we draw a random sample from that population. In this case, essentially no statistical knowledge is needed for one to imagine that our best "guess" (i.e. estimate) for the population mean will be the sample average, that is, the arithmetic mean of the data collected. The theoretical justification for this (intuitively appealing) guess is provided by the results of Chapter 8, with the use of the so-called laws of large numbers. The central limit theorem, one of the fundamental results in probability theory, is also presented in the same chapter; it goes further to consider the difference between the sample average and the theoretical mean and highlights the importance of the normal distribution in statistical theory and practice.

This book is intended as a textbook for a...