Title:
Probability : the science of uncertainty with applications to investments, insurance, and engineering
Personal Author:
Publication Information:
Pacific Grove, CA: Brooks/Cole, 2001
ISBN:
9780534366032
Subject Term:
Available:*
Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000004552851 | QA273 B37 2001 | Open Access Book | Book | Searching... |
Searching... | 30000004500728 | QA273 B37 2001 | Open Access Book | Book | Searching... |
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Summary
Summary
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
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 |