# Book:Michael A. Bean/Probability: The Science of Uncertainty

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## Michael A. Bean:

## Michael A. Bean: *Probability: The Science of Uncertainty*

Published $2001$, **Thomson Brooks/Cole**

- ISBN 978-0-8218-4792-3.

### Subject Matter

### Contents

- 1 Introduction
- 1.1 What Is Probability?
- 1.2 How Is Uncertainty Quantified?
- 1.3 Probability in Engineering and the Sciences
- 1.4 What Is Actuarial Science?
- 1.5 What Is Financial Engineering?
- 1.6 Interpretations of Probability
- 1.7 Probability Modeling in Practice
- 1.8 Outline of This Book
- 1.9 Chapter Summary
- 1.10 Further Reading
- 1.11 Exercises

- 2 A Survey of Some Basic Concepts Through Examples
- 2.1 Payoff in a Simple Game
- 2.2 Choosing Between Payoffs
- 2.3 Future Lifetimes
- 2.4 Simple and Compound Growth
- 2.5 Chapter Summary
- 2.6 Exercises

- 3 Classical Probability
- 3.1 The Formal Language of Classical Probability
- 3.2 Conditional Probability
- 3.3 The Law of Total Probability
- 3.4 Bayes' Theorem
- 3.5 Chapter Summary
- 3.6 Exercises
- 3.7 Appendix on Sets, Combinatorics, and Basic Probability Rules

- 4 Random Variables and Probability Distributions
- 4.1 Definitions and Basic Properties
- 4.1.1 What Is a Random Variable?
- 4.1.2 What Is a Probability Distribution?
- 4.1.3 Types of Distributions
- 4.1.4 Probability Mass Functions
- 4.1.5 Probability Density Functions
- 4.1.6 Mixed Distributions
- 4.1.7 Equality and Equivalence of Random Variables
- 4.1.8 Random Vectors and Bivariate Distribution
- 4.1.9 Dependence and Independence of Random Variables
- 4.1.10 The Law of Total Probability and Bayes' Theorem (Distributional Forms)
- 4.1.11 Arithmetic Operations on Random Variables
- 4.1.12 The Difference Between Sums and Mixtures
- 4.1.13 Exercises

- 4.2 Statistical Measures of Expectation, Variation, and Risk
- 4.2.1 Expectation
- 4.2.2 Deviation from Expectation
- 4.2.3 Higher Moments
- 4.2.4 Exercises

- 4.3 Alternative Ways of Specifying Probability Distributions
- 4.3.1 Moment and Cumulant Generating Functions
- 4.3.2 Survival and Hazard Functions
- 4.3.3 Exercises

- 4.4 Chapter Summary
- 4.5 Additional Exercises
- 4.6 Appendix on Generalized Density Functions (Optional)

- 4.1 Definitions and Basic Properties

- 5 Special Discrete Distributions
- 5.1 The Binomial Distribution
- 5.2 The Poisson Distribution
- 5.3 The Negative Binomial Distribution
- 5.4 The Geometric Distribution
- 5.5 Exercises

- 6 Special Continuous Distributions
- 6.1 Special Continuous Distributions for Modeling Uncertain Sizes
- 6.1.1 The Exponential Distribution
- 6.1.2 The Gamma Distribution
- 6.1.3 The Pareto Distribution

- 6.2 Special Continuous Distributions for Modeling Lifetimes
- 6.2.1 The Weibull Distribution
- 6.2.2 The DeMoivre Distribution

- 6.3 Other Special Distributions
- 6.3.1 The Normal Distribution
- 6.3.2 The Lognormal Distribution
- 6.3.3 The Beta Distribution

- 6.4 Exercises

- 6.1 Special Continuous Distributions for Modeling Uncertain Sizes

- 7 Transformations of Random Variables
- 7.1 Determining the Distribution of a Transformed Random Variable
- 7.2 Expectation of a Transformed Random Variable
- 7.3 Insurance Contracts with Caps, Deductibles, and Coinsurance (Optional)
- 7.4 Life Insurance and Annuity Contracts (Optional)
- 7.5 Reliability of Systems with Multiple Components or Processes (Optional)
- 7.6 Trigonometric Transformations (Optional)
- 7.7 Exercises

- 8 Sums and Products of Random Variables
- 8.1 Techniques for Calculating the Distribution of a Sum
- 8.1.1 Using the Joint Density
- 8.1.2 Using the Law of Total Probability
- 8.1.3 Convolutions

- 8.2 Distributions of Products and Quotients
- 8.3 Expectations of Sums and Products
- 8.3.1 Formulas for the Expectation of a Sum or Product
- 8.3.2 The Cauchy-Schwarz Inequality
- 8.3.3 Covariance and Correlation

- 8.4 The Law of Large Numbers
- 8.4.1 Motivating Example: Premium Determination in Insurance
- 8.4.2 Statement and Proof of the Law
- 8.4.3 Some Misconceptions Surrounding the Law of Large Numbers

- 8.5 The Central Limit Theorem
- 8.6 Normal Power Approximations (Optional)
- 8.7 Exercises

- 8.1 Techniques for Calculating the Distribution of a Sum

- 9 Mixtures and Compound Distributions
- 9.1 Definitions and Basic Properties
- 9.2 Some Important Exampels of Mixtures Arising in Insurance
- 9.3 Mean and Variance of a Mixture
- 9.4 Moment Generating Functions of a Mixture
- 9.5 Compound Distributions
- 9.5.1 General Formulas
- 9.5.2 Special Compound Distributions

- 9.6 Exercises

- 10 The Markowitz Investment Portfolio Selection Model
- 10.1 Portfolios of Two Securities
- 10.2 Portfolios of Two Risky Securities and a Risk-Free Asset
- 10.3 Portfolio Selection with Many Securities
- 10.4 The Capital Asset Pricing Model
- 10.5 Further Reading
- 10.6 Exercises

- Appendixes
- A The Gamma Function
- B The Incomplete Gamma Function
- C The Beta Function
- D The Incomplete Beta Function
- E The Standard Normal Distribution
- F
*Mathematica*Commands for Generating the Graphs of Special Distributions - G Elementary Financial Mathematics

- Answers to Selected Exercises
- Index