Probability: the science of uncertainty with applications to investments, insurance and engineering

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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