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

Subject Matter

 * Probability Theory

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)


 * 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


 * 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


 * 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