Enhanced descriptions from Syndetics:

This textbook for a one-semester course in probability covers combinatorial probability theory based on sets and counting, random variables and probability distribution, special discrete and continuous distributions, and transformations of random variables. A separate chapter provides four extended examples that apply many of the key concepts. Anno

Table of contents provided by Syndetics

• 1 Introduction (p. 1)
• 1.1 What Is Probability? (p. 1)
• 1.2 How Is Uncertainty Quantified? (p. 2)
• 1.3 Probability in Engineering and the Sciences (p. 5)
• 1.4 What Is Actuarial Science? (p. 6)
• 1.5 What Is Financial Engineering? (p. 9)
• 1.6 Interpretations of Probability (p. 11)
• 1.7 Probability Modeling in Practice (p. 13)
• 1.8 Outline of This Book (p. 14)
• 1.9 Chapter Summary (p. 15)
• 1.10 Further Reading (p. 16)
• 1.11 Exercises (p. 17)
• 2 A Survey of Some Basic Concepts Through Examples (p. 19)
• 2.1 Payoff in a Simple Game (p. 19)
• 2.2 Choosing Between Payoffs (p. 25)
• 2.3 Future Lifetimes (p. 36)
• 2.4 Simple and Compound Growth (p. 42)
• 2.5 Chapter Summary (p. 49)
• 2.6 Exercises (p. 51)
• 3 Classical Probability (p. 57)
• 3.1 The Formal Language of Classical Probability (p. 58)
• 3.2 Conditional Probability (p. 64)
• 3.3 The Law of Total Probability (p. 68)
• 3.4 Bayes' Theorem (p. 72)
• 3.5 Chapter Summary (p. 75)
• 3.6 Exercises (p. 76)
• 3.7 Appendix on Sets, Combinatorics, and Basic Probability Rules (p. 85)
• 4 Random Variables and Probability Distributions (p. 91)
• 4.1 Definitions and Basic Properties (p. 91)
• 4.1.1 What Is a Random Variable? (p. 91)
• 4.1.2 What Is a Probability Distribution? (p. 92)
• 4.1.3 Types of Distributions (p. 94)
• 4.1.4 Probability Mass Functions (p. 97)
• 4.1.5 Probability Density Functions (p. 97)
• 4.1.6 Mixed Distributions (p. 100)
• 4.1.7 Equality and Equivalence of Random Variables (p. 102)
• 4.1.8 Random Vectors and Bivariate Distributions (p. 104)
• 4.1.9 Dependence and Independence of Random Variables (p. 113)
• 4.1.10 The Law of Total Probability and Bayes' Theorem (Distributional Forms) (p. 119)
• 4.1.11 Arithmetic Operations on Random Variables (p. 124)
• 4.1.12 The Difference Between Sums and Mixtures (p. 125)
• 4.1.13 Exercises (p. 126)
• 4.2 Statistical Measures of Expectation, Variation, and Risk (p. 130)
• 4.2.1 Expectation (p. 130)
• 4.2.2 Deviation from Expectation (p. 143)
• 4.2.3 Higher Moments (p. 149)
• 4.2.4 Exercises (p. 153)
• 4.3 Alternative Ways of Specifying Probability Distributions (p. 155)
• 4.3.1 Moment and Cumulant Generating Functions (p. 155)
• 4.3.2 Survival and Hazard Functions (p. 167)
• 4.3.3 Exercises (p. 170)
• 4.4 Chapter Summary (p. 173)
• 4.5 Additional Exercises (p. 177)
• 4.6 Appendix on Generalized Density Functions (Optional) (p. 178)
• 5 Special Discrete Distributions (p. 186)
• 5.1 The Binomial Distribution (p. 187)
• 5.2 The Poisson Distribution (p. 195)
• 5.3 The Negative Binomial Distribution (p. 200)
• 5.4 The Geometric Distribution (p. 206)
• 5.5 Exercises (p. 209)
• 6 Special Continuous Distributions (p. 221)
• 6.1 Special Continuous Distributions for Modeling Uncertain Sizes (p. 221)
• 6.1.1 The Exponential Distribution (p. 221)
• 6.1.2 The Gamma Distribution (p. 226)
• 6.1.3 The Pareto Distribution (p. 233)
• 6.2 Special Continuous Distributions for Modeling Lifetimes (p. 235)
• 6.2.1 The Weibull Distribution (p. 235)
• 6.2.2 The DeMoivre Distribution (p. 241)
• 6.3 Other Special Distributions (p. 245)
• 6.3.1 The Normal Distribution (p. 245)
• 6.3.2 The Lognormal Distribution (p. 256)
• 6.3.3 The Beta Distribution (p. 260)
• 6.4 Exercises (p. 265)
• 7 Transformations of Random Variables (p. 280)
• 7.1 Determining the Distribution of a Transformed Random Variable (p. 281)
• 7.2 Expectation of a Transformed Random Variable (p. 289)
• 7.3 Insurance Contracts with Caps, Deductibles, and Coinsurance (Optional) (p. 297)
• 7.4 Life Insurance and Annuity Contracts (Optional) (p. 303)
• 7.5 Reliability of Systems with Multiple Components or Processes (Optional) (p. 311)
• 7.6 Trigonometric Transformations (Optional) (p. 317)
• 7.7 Exercises (p. 319)
• 8 Sums and Products of Random Variables (p. 325)
• 8.1 Techniques for Calculating the Distribution of a Sum (p. 325)
• 8.1.1 Using the Joint Density (p. 326)
• 8.1.2 Using the Law of Total Probability (p. 331)
• 8.1.3 Convolutions (p. 336)
• 8.2 Distributions of Products and Quotients (p. 337)
• 8.3 Expectations of Sums and Products (p. 339)
• 8.3.1 Formulas for the Expectation of a Sum or Product (p. 339)
• 8.3.2 The Cauchy-Schwarz Inequality (p. 340)
• 8.3.3 Covariance and Correlation (p. 341)
• 8.4 The Law of Large Numbers (p. 345)
• 8.4.1 Motivating Example: Premium Determination in Insurance (p. 346)
• 8.4.2 Statement and Proof of the Law (p. 349)
• 8.4.3 Some Misconceptions Surrounding the Law of Large Numbers (p. 351)
• 8.5 The Central Limit Theorem (p. 352)
• 8.6 Normal Power Approximations (Optional) (p. 354)
• 8.7 Exercises (p. 356)
• 9 Mixtures and Compound Distributions (p. 363)
• 9.1 Definitions and Basic Properties (p. 363)
• 9.2 Some Important Examples of Mixtures Arising in Insurance (p. 366)
• 9.3 Mean and Variance of a Mixture (p. 373)
• 9.4 Moment Generating Function of a Mixture (p. 378)
• 9.5 Compound Distributions (p. 379)
• 9.5.1 General Formulas (p. 380)
• 9.5.2 Special Compound Distributions (p. 382)
• 9.6 Exercises (p. 384)
• 10 The Markowitz Investment Portfolio Selection Model (p. 396)
• 10.1 Portfolios of Two Securities (p. 397)
• 10.2 Portfolios of Two Risky Securities and a Risk-Free Asset (p. 403)
• 10.3 Portfolio Selection with Many Securities (p. 409)
• 10.4 The Capital Asset Pricing Model (p. 411)
• 10.5 Further Reading (p. 414)
• 10.6 Exercises (p. 415)
• Appendixes (p. 421)
• A The Gamma Function (p. 421)
• B The Incomplete Gamma Function (p. 423)
• C The Beta Function (p. 428)
• D The Incomplete Beta Function (p. 429)
• E The Standard Normal Distribution (p. 430)
• F Mathematica Commands for Generating the Graphs of Special Distributions (p. 432)
• G Elementary Financial Mathematics (p. 434)
• Answers to Selected Exercises (p. 437)
• Index (p. 441)