Fundamentals of Probability, with Stochastic Processes

by Ghahramani, Saeed

• ISBN13: 9780131453401
• ISBN10: 0131453408
• Edition: 3rd
• Format: Paperback
• Copyright: 2004-07-22
• Publisher: Pearson

Summary

Presenting probability in a natural way, this book uses interesting, carefully selected instructive examples that explain the theory, definitions, theorems, and methodology.Fundamentals of Probabilityhas been adopted by theAmerican Actuarial Societyas one of its main references for the mathematical foundations of actuarial science.Topics include: axioms of probability; combinatorial methods; conditional probability and independence; distribution functions and discrete random variables; special discrete distributions; continuous random variables; special continuous distributions; bivariate distributions; multivariate distributions; sums of independent random variables and limit theorems; stochastic processes; and simulation.For anyone employed in the actuarial division of insurance companies and banks, electrical engineers, financial consultants, and industrial engineers.

Table of Contents

(Note:Each chapter ends with a Review Problems section.)

Axioms of Probability

Combinatorial Methods

Conditional Probability and Independence

Distribution Functions and Discrete Random Variables

Special Discrete Distributions

Continuous Random Variables

Special Continuous Distributions

Bivariate Distributions

Multivariate Distributions

More Expectations and Variances

Sums of Independent Random Variables and Limit Theorems

Stochastic Processes

Simulation

Appendix Tables

Answers to Odd-Numbered Exercises

Index

Table of Contents provided by Publisher. All Rights Reserved.

Excerpts

This one- or two-term basic probability text is written for majors in mathematics, physical sciences, engineering, statistics, actuarial science, business and finance, operations research, and computer science. It can also be used by students who have completed a basic calculus course. Our aim is to present probability in a natural way: through interesting and instructive examples and exercises that motivate the theory, definitions, theorems, and methodology. Examples and exercises have been carefully designed to arouse curiosity and hence encourage the students to delve into the theory with enthusiasm.Authors are usually faced with two opposing impulses. One is a tendency to put too much into the book, becauseeverythingis important andeverythinghas to be said the author's way! On the other hand, authors must also keep in mind a clear definition of the focus, the level, and the audience for the book, thereby choosing carefully what should be "in" and what "out." Hopefully, this book is an acceptable resolution of the tension generated by these opposing forces.Instructors should enjoy the versatility of this text. They can choose their favorite problems and exercises from a collection of 1558 and, if necessary, omit some sections and/or theorems to teach at an appropriate level.Exercises for most sections are divided into two categories: A and B. Those in category A are routine, and those in category B are challenging. However, not all exercises in category B are uniformly challenging. Some of those exercises are included because students find them somewhat difficult.I have tried to maintain an approach that is mathematically rigorous and, at the same time, closely matches the historical development of probability. Whenever appropriate, I include historical remarks, and also include discussions of a number of probability problems published in recent years in journals such asMathematics MagazineandAmerican Mathematical Monthly.These are interesting and instructive problems that deserve discussion in classrooms.Chapter 13 concerns computer simulation. That chapter is divided into several sections, presenting algorithms that are used to find approximate solutions to complicated probabilistic problems. These sections can be discussed independently when relevant materials from earlier chapters are being taught, or they can be discussed concurrently, toward the end of the semester. Although I believe that the emphasis should remain on concepts, methodology, and the mathematics of the subject, I also think that students should be asked to read the material on simulation and perhaps do some projects. Computer simulation is an excellent means to acquire insight into the nature of a problem, its functions, its magnitude, and the characteristics of the solution. Other Continuing Features The historical roots and applications of many of the theorems and definitions are presented in detail, accompanied by suitable examples or counterexamples. As much as possible, examples and exercises for each section do not refer to exercises in other chapters or sections--a style that often frustrates students and instructors. Whenever a new concept is introduced, its relationship to preceding concepts and theorems is explained. Although the usual analytic proofs are given, simple probabilistic arguments are presented to promote deeper understanding of the subject. The book begins with discussions on probability and its definition, rather than with combinatorics. I believe that combinatorics should be taught after students have learned the preliminary concepts of probability. The advantage of this approach is that the need for methods of counting will occur naturally to students, and the connection between the two areas becomes clear from the beginning. Moreover, combinatorics becomes more interesting and enjoyable. Students