Probability and Statistical Models

by Gupta, Arjun K.; Zeng, Wei-Bin; Wu, Yanhong

• ISBN13: 9780817649869
• ISBN10: 0817649867
• Format: Hardcover
• Copyright: 2010-08-20
• Publisher: Birkhauser

Summary

With an emphasis on models and techniques, this textbook introduces many of the fundamental concepts of stochastic modeling that are now a vital component of almost every scientific investigation. These models form the basis of well-known parametric lifetime distributions such as exponential, Weibull, and gamma distributions, as well as change-point and mixture models. The authors also consider more general notions of non-parametric lifetime distribution classes. In particular, emphasis is placed on laying the foundation for solving problems in reliability, insurance, finance, and credit risk. Exercises and solutions to selected problems accompany each chapter in order to allow students to explore these foundations.The key subjects covered include:* Exponential distributions and the Poisson process* Parametric lifetime distributions* Non-parametric lifetime distribution classes* Multivariate exponential extensions* Association and dependence* Renewal theory* Problems in reliability, insurance, finance, and credit riskThis work differs from traditional probability textbooks in a number of ways. Since no measure theory knowledge is necessary to understand the material and coverage of the central limit theorem and normal theory related topics has been omitted, the work may be used as a single-semester senior undergraduate or first-year graduate textbook as well as in a second course on probability modeling. Many of the chapters that examine central topics in applied probability can be read independently, allowing both instructors and readers extra flexibility in their use of the book.Probability and Statistical Models is for a wide audience including advanced undergraduate and beginning-level graduate students, researchers, and practitioners in mathematics, statistics, engineering, and economics.

Table of Contents

Preliminaries	p. 1
Introduction	p. 1
Notations	p. 2
Random Variable and Distribution function	p. 4
Mean and Variance	p. 5
Joint and Conditional Distributions	p. 8
Joint Distribution	p. 8
Independent Sums and Laws	p. 9
Conditional Distribution and Mean	p. 10
Survival Function and Failure Rate	p. 13
Survival Function and Failure Rate	p. 13
Mean and Mean Residual Life	p. 15
Cauchy Functional Equation	p. 16
Problems	p. 17
Exponential Distribution	p. 23
Introduction	p. 23
Exponential Distribution	p. 23
Characterization of Exponential Distribution	p. 27
Memoryless Property	p. 27
Constant Failure Rate Function	p. 30
Extreme Value Distribution	p. 30
Order Statistics and Exponential Distribution	p. 32
Some Properties of Order Statistics	p. 32
Characterization Based on Order Statistics	p. 35
Record Values	p. 37
More Applications	p. 37
Problems	p. 40
Poisson Process	p. 45
Poisson Process as a Counting Process	p. 45
Characterization of Poisson Processes as Counting Processes	p. 47
Poisson Process as a Renewal Process	p. 53
Further Properties of Poisson Process	p. 57
Superposition Process	p. 57
Decomposition of Poisson Process	p. 58
Examples of Poisson Process	p. 60
Problems	p. 67
Parametric Families of Lifetime Distributions	p. 71
Weibull Distribution	p. 71
Gamma Distribution	p. 74
Change-Point Model	p. 78
Mixture Exponential Distribution	p. 79
IFR (DFR) and Mixture Erlang Distribution	p. 81
Problems	p. 84
Lifetime Distribution Classes	p. 87
IFR and DFR	p. 87
IFR and PF2	p. 87
Smoothness of IFR Distribution	p. 90
A Sufficient Condition	p. 91
IFRA and DFRA Classes	p. 92
Several Lifetime Distribution Classes	p. 95
Preservation of Lifetime Distributions Under Reliability Operations	p. 99
Independent Sums	p. 99
Mixture of Lifetime Distributions	p. 101
Shock Models and Lifetime Distribution Classes	p. 104
IFRA Property of Shock Model	p. 104
Extension of Cumulative Damage Model	p. 107
General Cumulative Damage Model	p. 108
Shock Models Leading to Other Lifetime Distributions	p. 110
Problems	p. 112
Multivariate Lifetime Distributions	p. 117
Basic Properties of Bivariate Distributions	p. 117
Bivariate Memoryless Property	p. 120
Property of the BVE	p. 125
A Nonfatal Shock Model	p. 133
Absolutely Continuous Bivariate Exponential Extensions	p. 135
Problems	p. 139
Association and Dependence	p. 141
Several Concepts of Association	p. 141
MTP2 Distribution	p. 146
Multivariate Failure Rate and Distribution class	p. 149
Negative Association	p. 151
Problems	p. 156
Renewal Theory	p. 159
Renewal Theorem	p. 159
High-Order Approximations and Bounds	p. 163
Delayed Renewal Process	p. 166
Defective Renewal Process	p. 169
Problems	p. 175
Risk Theory	p. 179
Classical Risk Model	p. 179
Approximation and Bounds for Ruin Probability	p. 181
Deficit at Ruin	p. 183
Large Claim Case	p. 185
Bounds in terms of NWU (NBU) Distribution Classes	p. 186
Subexponential Classes	p. 190
Risk Sharing and Stop-Loss Reinsurance	p. 193
Problems	p. 196
Asset Pricing Theory	p. 199
Utility, Risk, and Pricing Kernel	p. 199
Utility and Risk	p. 199
Asset Pricing Formula and Pricing Kernel	p. 200
Models for Returns	p. 203
ß-Representation	p. 203
Frontier Expression	p. 204
Log-Normal Model	p. 204
Examples of Risk Assets	p. 205
Risk-Neutral Probabilties	p. 207
Option Pricing for Binomial Model	p. 208
Pricing Formula for Multiple Stages	p. 208
Binomial Model	p. 208
Portfolio Management	p. 210
Discrete Financial Market	p. 210
Risk Management	p. 211
Hedging Options	p. 213
Black-Scholes Formula	p. 216
Problems	p. 218
Credit Risk Modeling	p. 221
Two Models for Default Probability	p. 221
Basic Notation	p. 221
Reduced Form	p. 222
Structural Model	p. 224
Valuation of Default Risk	p. 225
No Recovery Zero-Coupon Defaultable Bond	p. 226
Non-Zero Recovery	p. 226
Actual and Risk Neutral Default Intensity	p. 227
Credit Rating: Default and Transition	p. 227
Credit Rating	p. 227
Rating Assignment	p. 229
Rating Transition	p. 229
Correlated Defaults	p. 230
Credit Metrics	p. 230
Correlated Default Intensities	p. 231
Copula-Based Correlation Modeling	p. 231
Credit Derivatives	p. 232
Credit Default Swaps	p. 233
Collateral Debt Obligations	p. 234
Problems	p. 235
Bibliographical Notes and Further Reading	p. 237
References	p. 241
Answers and Solutions to Selected Problems	p. 245
Index	p. 265
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