**Pure and Applied Undergraduate Texts**

Volume: 6;
2001;
448 pp;
Hardcover

MSC: Primary 60;

Print ISBN: 978-0-8218-4792-3

Product Code: AMSTEXT/6

List Price: $76.00

Individual Member Price: $60.80

**Electronic ISBN: 978-1-4704-1119-0
Product Code: AMSTEXT/6.E**

List Price: $76.00

Individual Member Price: $60.80

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#### Supplemental Materials

# Probability: The Science of Uncertainty: with Applications to Investments, Insurance, and Engineering

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*Michael A. Bean*

This book covers the basic probability of distributions with an emphasis on applications from the areas of investments, insurance, and engineering. Written by a Fellow of the Casualty Actuarial Society and the Society of Actuaries with many years of experience as a university professor and industry practitioner, the book is suitable as a text for senior undergraduate and beginning graduate students in mathematics, statistics, actuarial science, finance, or engineering as well as a reference for practitioners in these fields. The book is particularly well suited for students preparing for professional exams, and for several years it has been recommended as a textbook on the syllabus of examinations for the Casualty Actuarial Society and the Society of Actuaries.

In addition to covering the standard topics and probability distributions, this book includes separate sections on more specialized topics such as mixtures and compound distributions, distributions of transformations, and the application of specialized distributions such as the Pareto, beta, and Weibull. The book also has a number of unique features such as a detailed description of the celebrated Markowitz investment portfolio selection model. A separate section contains information on how graphs of the specific distributions studied in the book can be created using Mathematica™.

The book includes a large number of problems of varying difficulty. A student manual with solutions to selected problems is available electronically from the "Solutions Manual" link above.

An instructor's manual for this title is available electronically. Please send email to textbooks@ams.org for more information.

#### Table of Contents

# Table of Contents

## Probability: The Science of Uncertainty: with Applications to Investments, Insurance, and Engineering

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- About the Author iii4 free
- Preface v6 free
- Contents ix10 free
- Chapter 1. Introduction 116 free
- 1.1 What Is Probability? 116
- 1.2 How Is Uncertainty Quantified? 217
- 1.3 Probability in Engineering and the Sciences 520
- 1.4 What Is Actuarial Science? 621
- 1.5 What Is Financial Engineering? 924
- 1.6 Interpretations of Probability 1126
- 1.7 Probability Modeling in Practice 1328
- 1.8 Outline of This Book 1429
- 1.9 Chapter Summary 1530
- 1.10 Further Reading 1631
- 1.11 Exercises 1732

- Chapter 2. A Survey of Some Basic Concepts Through Examples 1934
- Chapter 3. Classical Probability 5772
- Chapter 4. Random Variables and Probability Distributions 91106
- 4.1 Definitions and Basic Properties 91106
- 4.1.1 What Is a Random Variable? 91106
- 4.1.2 What Is a Probability Distribution? 92107
- 4.1.3 Types of Distributions 94109
- 4.1.4 Probability Mass Functions 97112
- 4.1.5 Probability Density Functions 97112
- 4.1.6 Mixed Distributions 100115
- 4.1.7 Equality and Equivalence of Random Variables 102117
- 4.1.8 Random Vectors and Bivariate Distributions 104119
- 4.1.9 Dependence and Independence of Random Variables 113128
- 4.1.10 The Law of Total Probability and Bayes' Theorem (Distributional Forms) 119134
- 4.1.11 Arithmetic Operations on Random Variables 124139
- 4.1.12 The Difference Between Sums and Mixtures 125140
- 4.1.13 Exercises 126141

- 4.2 Statistical Measures of Expectation, Variation, and Risk 130145
- 4.3 Alternative Ways of Specifying Probability Distributions 155170
- 4.4 Chapter Summary 173188
- 4.5 Additional Exercises 177192
- 4.6 Appendix on Generalized Density Functions (Optional) 178193

- Chapter 5. Special Discrete Distributions 186201
- Chapter 6. Special Continuous Distributions 221236
- Chapter 7. Transformations of Random Variables 280295
- 7.1 Determining the Distribution of a Transformed Random Variable 281296
- 7.2 Expectation of a Transformed Random Variable 289304
- 7.3 Insurance Contracts with Caps, Deductibles, and Coinsurance (Optional) 297312
- 7.4 Life Insurance and Annuity Contracts (Optional) 303318
- 7.5 Reliability of Systems with Multiple Components or Processes (Optional) 311326
- 7.6 Trigonometric Transformations (Optional) 317332
- 7.7 Exercises 319334

- Chapter 8. Sums and Products of Random Variables 325340
- 8.1 Techniques for Calculating the Distribution of a Sum 325340
- 8.2 Distributions of Products and Quotients 337352
- 8.3 Expectations of Sums and Products 339354
- 8.4 The Law of Large Numbers 345360
- 8.5 The Central Limit Theorem 352367
- 8.6 Normal Power Approximations (Optional) 354369
- 8.7 Exercises 356371

- Chapter 9. Mixtures and Compound Distributions 363378
- Chapter 10. The Markowitz Investment Portfolio Selection Model 396411
- Appendixes 421436
- A: The Gamma Function 421436
- B: The Incomplete Gamma Function 423438
- C: The Beta Function 428443
- D: The Incomplete Beta Function 429444
- E: The Standard Normal Distribution 430445
- F: Mathematica Commands for Generating the Graphs of Special Distributions 432447
- G: Elementary Financial Mathematics 434449

- Answers to Selected Exercises 437452
- Index 441456
- Back Cover Back Cover1464

#### Readership

Undergraduate and graduate students in mathematics, finance, or engineering interested in probability and its applications.

- Introduction
- A survey of some basic concepts through examples
- Classical probability
- Random variables and probability distributions
- Special discrete distributions
- Special continuous distributions
- Transformations of random variables
- Sums and products of random variables
- Mixtures and compound distributions
- The Markowitz investment portfolio selection model
- Appendixes
- Answers to selected exercises
- Index