本书是为投资、保险和工程等应用专业编写的概率论教材。作者既向读者介绍了必需的概率论知识，又突出了其他书籍中未详细介绍的独具实用价值的内容，如条件概率、贝叶斯定理、混合概率分布、Markowitz投资选择模型等。全书贯穿以投资、保险和其他工程中的应用问题，饮食大量的例题和习题，并把不确定性科学范畴的要领和方法与金融领域的实践紧密结合起来。
本书涉及的题材广泛，可作为运筹学、统计学、精算科学、管理科学和决策科学等专业概率论课程的教材或教学参考书，也适合学生或专业人员自学。
The idea to write this book first came to me in the spring of l995, shortly after I joined the faculty of the University of Michigan. At that time, a major review of the entire undergraduate curriculum was underway, the purpose of which was to ensure that undergraduate education remain relevant in the face of a rapidly changing word. About the same time, the Society of Actuaries, which oversees the education of actuaries in North America through the administration of its Professional examinations, embarked on a major redesign of its own curriculum to keep abreast of the extraordinary changes taking place in the financial services industry This time also saw the emergence of financial engineering as a new profession and the rise of programs in financial engineering and financial mathematics around the worid. My goal in writing this book was to update the undergraduate probability curriculum to reflect these changes and to incorporate many of the new and interesting applications of probability arising in the fields of engineering, insurance, and invesiments.
Key Features of This Text
This book has several features that dishnguish it from other probability texts currently on the market:
Key concepts are introduced through detailed motivating examples.
Random variables and probability distributions are introduced early in the text.
The text has a large number of detailed worked-out examples and problems with an emphasis on applications from engineering, insurance, and investments.
There is a wide range of exercises of varying difficulty, many of which are suitable for student projects or group work.
The text includes topics not covered or not emphasized in other probability texts, such as the geometric expected value, normal power approximations, mixtures, and portfolio selection models.
The text is written in a cleap concise, expository style, with extensive graphical illustrations throughout, making it well suited for individual study or self-learning.
How to Use This Book
This book can be used in a variety of probability courses with a variety of teaching styles. There is considerably more material in this book than would normally be covered in a onesemester course. Hence, an instructor will have to be selective in what is covered.
What I consider to be core material for an undergraduate probability course is contained in Chapters 3, 4, 5, and 6. An instructor teaching probability should plan on covering most of the material in these chapters, although discussions of some specialized topics such as the pareto distribution and the beta distribution can be omitted without loss of continuity.
The material in Chapter 7 and Chapter 8 is also important and should be covered to some extent. Instructors teaching engineering students will probably want to discuss the techniques for determining the distribution of a transformed random variable and the distributions of sums and products (7. l and 8. 1) quite thoroughly. Instructors teaching other types of students may wish to focus on the law of large numbers (8.4) instead. The sections labeled as being "optional" may be omitted without loss of continuity.
Chapter 2 is unique in that it uses four extended examples to motivate many of the key concepts covered in the rest of the book. I have found that by discussing these examples at the beginning of the course (i.e., before covering Chapters 3 through 8), students are able to make important conceptual discoveries early on and end up learning a great deal of probability theory in a relatively short period of time. Instructors familiar with the discovery method of learning should be quite comfortable using Chapter 2 in this way. Instructors accustomed to teaching in a more tradtional way can begin the course at Chapter 3 (after a brief survey of Chapter l) and use Chapter 2 selectively or omit it entirely.
The material in Chapters 9 and l0 is supplementary and would not normally be covered in a one-semester course in probability. Howevef, this material is good for student projects.
Chapter summaries are provided in the first four chapters to recap the main ideas and help the reader acquire perspective on the subject. Chapters 5 and 6 are written in a summary style throughout and hence do not require sepot sununary sections. Chapters 7 through l0 are designed to be covered selectively and do not contain summary sections.
An instructor's manual with solutions to all of the exercises in the book is available with a bound-in CD. This manual contains a wealth of material including detailed descriptions of the Mathematica commands for constructing the graphs in this book. It is freely available to insmictors who adopt this book as a text for their course. For details on how to obtain a copy, contact your Brooks/Cole representative or visit the Brooks/Cole Web site at www.brookscole.com.
Acknowledgments
Writing a textbook of this magnitllde is a major undertaking which requires the assistance of many people. I would like to begin by thanking my pubisher, Gary W. Ostedt, for agreeing to take on this project and by thanking Carol Benedict, Kelsey McGee, Karin Sandberg, Dan Thiem, and the rest of the Brooks/Cole team for their part in making this book a reality Thanks also go to Robin Gold at Forbes Mill Dess for keeping production on track under a tight schedule and for being patient with me when other responsibilihes demanded my attention.
I would also like to thank the reviewers for their valuable commnts and suggestions, many of which have been incorporated into the final manuscript. These reviewers inchid Phillip Beckwith of Michigan Technological University, John Holcomb of Cleveland State University, Paul Holmes of Clemson University, Ian McKeague of Florida State University, and Harry Panjer of the University of Waterloo, former president of the Canadian Institute of Actuaries.
Special thanks also go to my colleague Jack Goldberg for his helpful advice on the publication process (from a textbook author's perspective) and to John Birse for supporting this project in its early stages. I am also grateful to the National Science Foundation and the Center for Research on Learning and Teaching at the University of Michigan for their support of the cuniculum development initiahves that ultimately led to my writing this book. Finally, I would like to thank my parents for instilling in the an appreciation of the importance of education and for supporting me in all my endeavors.
Michael Bean
1 Introduction
1.1 What Is Protrability 1
1.2 How Is Uncertainty Quantified 2
1.3 Probability in Engineering and the Sciences 5
1.4 What Is Actuarial Science 6
1.5 What Is Financial Engineering 9
1.6 Interpretations of Probability 11
1.7 Probability Modeling in Practice 13
1.8 Outline of This Book 14
1.9 Chapter Summary 15
1.10 Further Reading 16
1.11 Exercises 17
2 A Survey of Some Basic Concepts Through Examples
2.1 Payoff in a Simple Game 19
2.2 Choosing Between Payoffs 25
2.3 Future Lifetimes 36
2.4 Simple and Compound Growth 42
2.5 Chapter Summary 49
2.6 Exercises 51
3 Classical Probability
3.1 The Formal Language of Classical Probability 58
3.2 Conditional Probability 64
3.3 The Law of Total Probability 68
3.4 Bayes' Theorem 72
3.5 Chapter Summary 75
3.6 Exercises 76
3.7 Appendix on Sets, Combinatorics, and Basic
4 Probability Rules 85
4 Random Variables and Probability Distributions
4.l Definitions and Basic Properties 91
4.1.1 What Is a Random Variable 91
4.1.2 What Is a Probability Distribution 92
4.1.3 Types of Distributions 94
4.1.4 Probability Mass Functions 97
4.1.5 Probability DensityFunctions 97
4.1.6 Mixed Distributions 100
4.1.7 Equality and Equivalence of Random Variables 102
4.1.8 Random Vectors and Bivariate Distributions 104
4.1.9 Dependence and Independence of Random Variables 113
4.1.10 The Law of Total Probability and Bayes' Theorem(Distributional Forms) 119
4.1.11 Arithmetic OPerations on Random Variables 124
4.1.12 The Difference Between Sums and Mxtores 125
4.1.13 Exercises 126
4.2 Statistical Measures of Expectation,Variation,and Risk 13O
4.2.1 Expectation 130
4.2.2 Deviation from Expectation 143
4.2.3 Higher Moments 149
4.2.4 Exercises 153
4.3 Alternative Ways of Specifying Probability Distributions 155
4.3.1 Moment and Cumulant Generating Functions 155
4.3.2 Survival and Hazard Functions 167
4.3.3 Exercises 170
4.4 Chapter Summary 173
4.5 Additional Exercises 177
4.6 Appendix on Generalized Density Functions (Optional)
5 Special Discrete Distributions
5.l The Binomial Distribution 187
5.2 The Poisson Distribution
5.3 The Negative Binomial Distribution 200
5.4 The Geometric Distribution 206
5.5 Exercises 209
6 Special Continuous Distributions
6.1 Special Continuous Distributions for Modeling Uncertain Sizes 221
6.1.1 TheExPonenhalDistribution 221
6.1.2 The Gamma Distribution 226
6.1.3 The Pareto Distribution 233
6.2 Special Continuous Distributions for Modeling Lifetimes
6.2.1 The Weibull Distribution 235
6.2.2 The DeMoivre Distribution 241
6.3 Other Special Distributions 245
6.3.1 The Normal Distribution 245
6.3.2 The Lognormal Distribution 256
6.3.3 The Beta Distribution 260
6.4 Exercises 265
7 Transformations of Random Variables
7.1 Determining the Distribution of a Transformed
Random Variable 281
7.2 Expectation of a Transformed Random Variable 289
7.3 Insurance Contracts with Caps, Deductibles,and Coinsurance (Optional) 297
7.4 Life Insurance and Annuity Contracts (Optional) 303
7.5 Reliability of Systems with Multiple Components
or Processes (Optional) 3l1
7.6 Trigonometric Transformations (Optional) 317
7.7 Exercises 3l9
8 Sums and Products of Random Variables
8.1 Techniques for Calculating the Distribution of a Sum
8.1.1 Using the Joint Density 326
8.1.2 Using tne Law or Thtal Probability 331
8.1.3 Convolutions 336
8.2 Distributions of Products and Quotients 337
8.3 Expectations of Sums and Products 339
8.3.1 Formulas for the Expectation of a Sum or Product 339
8.3.2 The Cauchy-Schwarz Inequality 34O
8.3.3 Covariance and Cormlation 341
8.4 The Law of Large Numbers 345
8.4.1 Motivating Example: Prmium Determination in Insurance
8.4.2 Statement and Proof of the Law
8.4.3 Some Misconceptions Surrounding the Law of Large Number
8.5 The Central Limit Theorem 3S2
8.6 Normal Power Approximations (Optional) 354
8.7 Exercises 356
9 Mixtures and Compound Distributions
9.1 Definitions and Basic Properties 363
9.2 Some Important Examples of Mixtures Arising in Insurance 366
9.3 Mean and Variance of a Mixture 373
9.4 Moment Generating Function of a Mixture 378
9.5 Compound Distributions 379
9.5.1 General Formulas 380
9.5.2 Special Compound Distributions 382
9.6 Exercises 384
10 The Markowitz Invetrient Portfolio Selection Model
l0.l Portfolios of Two Securities 397
10.2 Portfolios of Two Risky Securities and a Risk-Free Asset
l0.3 Portfolio Selection with Many Securities 409
l0.4 The Capital Asset Pricing Model 411
10.5 Further Reading 414
10.6 Exercises 415
Appendixes
A The Gamma Function 421
B The Incomplete Gamma Function 423
C The Beta Function 428
D The Incomplete Beta Function 429
E The Standard Normal Distribution 430
F Mathematica Commands for Generating the Graphs of Special Distributions 432
G Elementary Financial Mathematics 434
Answers to Selected Exercises 437
Index 441
