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 [email protected] for more information.
Michael A. Bean
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 Exercises
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
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
Random Variables and Probability Distributions 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 Statistical Measures of Expectation, Variation, and Risk Expectation Deviation from Expectation Higher Moments Exercises Alternative Ways of Specifying Probability Distributions Moment and Cumulant Generating Functions Survival and Hazard Functions Exercises Chapter Summary Additional Exercises Appendix on Generalized Density Functions (Optional)
Special Discrete Distributions The Binomial Distribution The Poisson Distribution The Negative Binomial Distribution The Geometric Distribution Exercises
Special Continuous Distributions Special Continuous Distributions for Modeling Uncertain Sizes The Exponential Distribution The Gamma Distribution The Pareto Distribution Special Continuous Distributions for Modeling Lifetimes The Weibull Distribution The DeMoivre Distribution Other Special Distributions The Normal Distribution The Lognormal Distribution The Beta Distribution Exercises
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
Sums and Products of Random Variables Techniques for Calculating the Distribution of a Sum Using the Joint Density Using the Law of Total Probability Convolutions Distributions of Products and Quotients Expectations of Sums and Products Formulas for the Expectation of a Sum or Product The Cauchy-Schwarz Inequality Covariance and Correlation The Law of Large Numbers Motivating Example: Premium Determination in Insurance &nbs