1 Introduction
1.1 What Is Protrability? 1
1.2 How Is Uncertainty Quantified? 2
1.3 Probability in Engineering and the Sciences 5
1.4 What Is Actuarial Science? 6
1.5 What Is Financial Engineering? 9
1.6 Interpretations of Probability 11
1.7 Probability Modeling in Practice 13
1.8 Outline of This Book 14
1.9 Chapter Summary 15
1.10 Further Reading 16
1.11 Exercises 17
2 A Survey of Some Basic Concepts Through Examples
2.1 Payoff in a Simple Game 19
2.2 Choosing Between Payoffs 25
2.3 Future Lifetimes 36
2.4 Simple and Compound Growth 42
2.5 Chapter Summary 49
2.6 Exercises 51
3 Classical Probability
3.1 The Formal Language of Classical Probability 58
3.2 Conditional Probability 64
3.3 The Law of Total Probability 68
3.4 Bayes' Theorem 72
3.5 Chapter Summary 75
3.6 Exercises 76
3.7 Appendix on Sets, Combinatorics, and Basic
4 Probability Rules 85
4 Random Variables and Probability Distributions
4.l Definitions and Basic Properties 91
4.1.1 What Is a Random Variable? 91
4.1.2 What Is a Probability Distribution? 92
4.1.3 Types of Distributions 94
4.1.4 Probability Mass Functions 97
4.1.5 Probability DensityFunctions 97
4.1.6 Mixed Distributions 100
4.1.7 Equality and Equivalence of Random Variables 102
4.1.8 Random Vectors and Bivariate Distributions 104
4.1.9 Dependence and Independence of Random Variables 113
4.1.10 The Law of Total Probability and Bayes' Theorem(Distributional Forms) 119
4.1.11 Arithmetic OPerations on Random Variables 124
4.1.12 The Difference Between Sums and Mxtores 125
4.1.13 Exercises 126
4.2 Statistical Measures of Expectation, Variation, and Risk 13O
4.2.1 Expectation 130
4.2.2 Deviation from Expectation 143
4.2.3 Higher Moments 149
4.2.4 Exercises 153
4.3 Alternative Ways of Specifying Probability Distributions 155
4.3.1 Moment and Cumulant Generating Functions 155
4.3.2 Survival and Hazard Functions 167
4.3.3 Exercises 170
4.4 Chapter Summary 173
4.5 Additional Exercises 177
4.6 Appendix on Generalized Density Functions (Optional)
5 Special Discrete Distributions
5.l The Binomial Distribution 187
5.2 The Poisson Distribution
5.3 The Negative Binomial Distribution 200
5.4 The Geometric Distribution 206
5.5 Exercises 209
6 Special Continuous Distributions
6.1 Special Continuous Distributions for Modeling Uncertain Sizes 221
6.1.1 TheExPonenhalDistribution 221
6.1.2 The Gamma Distribution 226
6.1.3 The Pareto Distribution 233
6.2 Special Continuous Distributions for Modeling Lifetimes
6.2.1 The Weibull Distribution 235
6.2.2 The DeMoivre Distribution 241
6.3 Other Special Distributions 245
6.3.1 The Normal Distribution 245
6.3.2 The Lognormal Distribution 256
6.3.3 The Beta Distribution 260
6.4 Exercises 265
7 Transformations of Random Variables
7.1 Determining the Distribution of a Transformed
Random Variable 281
7.2 Expectation of a Transformed Random Variable 289
7.3 Insurance Contracts with Caps, Deductibles,and Coinsurance (Optional) 297
7.4 Life Insurance and Annuity Contracts (Optional) 303
7.5 Reliability of Systems with Multiple Components
or Processes (Optional) 3l1
7.6 Trigonometric Transformations (Optional) 317
7.7 Exercises 3l9
8 Sums and Products of Random Variables
8.1 Techniques for Calculating the Distribution of a Sum
8.1.1 Using the Joint Density 326
8.1.2 Using tne Law or Thtal Probability 331
8.1.3 Convolutions 336
8.2 Distributions of Products and Quotients 337
8.3 Expectations of Sums and Products 339
8.3.1 Formulas for the Expectation of a Sum or Product 339
8.3.2 The Cauchy-Schwarz Inequality 34O
8.3.3 Covariance and Cormlation 341
8.4 The Law of Large Numbers 345
8.4.1 Motivating Example: Prmium Determination in Insurance
8.4.2 Statement and Proof of the Law
8.4.3 Some Misconceptions Surrounding the Law of Large Number
8.5 The Central Limit Theorem 3S2
8.6 Normal Power Approximations (Optional) 354
8.7 Exercises 356
9 Mixtures and Compound Distributions
9.1 Definitions and Basic Properties 363
9.2 Some Important Examples of Mixtures Arising in Insurance 366
9.3 Mean and Variance of a Mixture 373
9.4 Moment Generating Function of a Mixture 378
9.5 Compound Distributions 379
9.5.1 General Formulas 380
9.5.2 Special Compound Distributions 382
9.6 Exercises 384
10 The Markowitz Invetrient Portfolio Selection Model
l0.l Portfolios of Two Securities 397
10.2 Portfolios of Two Risky Securities and a Risk-Free Asset
l0.3 Portfolio Selection with Many Securities 409
l0.4 The Capital Asset Pricing Model 411
10.5 Further Reading 414
10.6 Exercises 415
Appendixes
A The Gamma Function 421
B The Incomplete Gamma Function 423
C The Beta Function 428
D The Incomplete Beta Function 429
E The Standard Normal Distribution 430
F Mathematica Commands for Generating the Graphs of Special Distributions 432
G Elementary Financial Mathematics 434
Answers to Selected Exercises 437
Index 441
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