**About the Book** 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 MathematicaTM.
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****1. Introduction**

- What Is Probability?
- How Is Uncertainty Quantified?
- Probability in Engineering and the Sciences
- What Is Actuarial Science?
- What Is Financial Engineering?
- Interpretations of Probability
- Probability Modeling in Practice
- Outline of This Book
- Chapter Summary
- Further Reading

**2. A Survey of Some Basic Concepts Through Examples**

- Payoff in a Simple Game
- Choosing Between Payoffs
- Future Lifetimes
- Simple and Compound Growth
- Chapter Summary
- Exercises

**3. Classical Probability**

- The Formal Language of Classical Probability
- Conditional Probability
- The Law of Total Probability
- Bayes' Theorem
- Chapter Summary
- Exercises
- Appendix on Sets, Combinatorics, and Basic Probability Rules

**4. Random Variables and Probability Distributions **

* **4.1 Definitions and Basic Properties*

- What Is a Random Variable?
- What Is a Probability Distribution?
- Types of Distributions
- Probability Mass Functions
- Probability Density Functions
- Mixed Distributions
- Equality and Equivalence of Random Variables
- Random Vectors and Bivariate Distributions
- Dependence and Independence of Random Variables
- The Law of Total Probability and Bayes' Theorem (Distributional Forms)
- Arithmetic Operations on Random Variables
- The Difference Between Sums and Mixtures
- Exercises

*4.2 Statistical Measures of Expectation, Variation, and Risk*

- Expectation
- Deviation from Expectation
- Higher Moments
- Exercises

* **4.3 Alternative Ways of Specifying Probability Distributions*

- Moment and Cumulant Generating Functions
- Survival and Hazard Functions
- Exercises

* 4.4 Chapter Summary*

* 4.5 Additional Exercises *

* 4.6 Appendix on Generalized Density Functions (Optional)*

**5. Special Discrete Distributions**

- The Binomial Distribution
- The Poisson Distribution
- The Negative Binomial Distribution
- The Geometric Distribution
- Exercises

**6. Special Continuous Distributions**

* **6.1 Special Continuous Distributions for Modeling
Uncertain Sizes*

- The Exponential Distribution
- The Gamma Distribution
- The Pareto Distribution

*6.2 Special Continuous Distributions for Modeling Lifetimes*

- The Weibull Distribution
- The DeMoivre Distribution

*6.3 Other Special Distributions*

- The Normal Distribution
- The Lognormal Distribution
- The Beta Distribution

*6.4 Exercises *

**7. Transformations of Random Variables**** **

- Determining the Distribution of a Transformed Random Variable
- Expectation of a Transformed Random Variable
- Insurance Contracts with Caps, Deductibles, and Coinsurance (Optional)
- Life Insurance and Annuity Contracts (Optional)
- Reliability of Systems with Multiple Components or Processes (Optional)
- Trigonometric Transformations (Optional)
- Exercises

**8. Sums and Products of Random Variables**

*8.1 Techniques for Calculating the Distribution of a Sum*

- Using the Joint Density
- Using the Law of Total Probability
- Convolutions

* **8.2 Distributions of Products and Quotients*

* 8.3 Expectations of Sums and Products*

- Formulas for the Expectation of a Sum or Product
- The Cauchy-Schwarz Inequality
- Covariance and Correlation

*8.4 The Law of Large Numbers*

- Motivating Example: Premium Determination in Insurance
- Statement and Proof of the Law
- Some Misconceptions Surrounding the Law of Large Numbers

* **8.5 The Central Limit Theorem*

* 8.6 Normal Power Approximations (Optional)*

* 8.7 Exercises*

**9. Mixtures and Compound Distributions**

- Definitions and Basic Properties
- Some Important Examples of Mixtures Arising in Insurance
- Mean and Variance of a Mixture
- Moment Generating Function of a Mixture
- Compound Distributions
- General Formulas
- Special Compound Distributions

- Exercises

**10. The Markowitz Investment Portfolio Selection Model**

- Portfolios of Two Securities
- Portfolios of Two Risky Securities and a Risk-Free Asset
- Portfolio Selection with Many Securities
- The Capital Asset Pricing Model
- Further Reading
- Exercises

**Appendixes**

- The Gamma Function
- The Incomplete Gamma Function
- The Beta Function
- The Incomplete Beta Function
- The Standard Normal Distribution
- Mathematica Commands for Generating the Graphs of Special Distributions
- Elementary Financial Mathematics

Answers to Selected Exercises

Index