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Probability and Statistical Models : Foundations for Problems in Reliability and Financial Mathematics - Arjun K. Gupta

Probability and Statistical Models

Foundations for Problems in Reliability and Financial Mathematics

Hardcover Published: 2nd September 2010
ISBN: 9780817649869
Number Of Pages: 267

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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 risk

This 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.

Industry Reviews

From the reviews:

"This is a nice introductory textbook on stochastic processes, basically devoted to the Poisson process and its variants. The basic results are well illustrated by many examples with many problems at the end of each chapter. ... The book is suitable for students that do not have an advanced training in the measure-theoretic aspects of probability or stochastic integration." (Henryk Gzyl, Zentralblatt MATH, Vol. 1215, 2011)

Preliminariesp. 1
Introductionp. 1
Notationsp. 2
Random Variable and Distribution functionp. 4
Mean and Variancep. 5
Joint and Conditional Distributionsp. 8
Joint Distributionp. 8
Independent Sums and Lawsp. 9
Conditional Distribution and Meanp. 10
Survival Function and Failure Ratep. 13
Survival Function and Failure Ratep. 13
Mean and Mean Residual Lifep. 15
Cauchy Functional Equationp. 16
Problemsp. 17
Exponential Distributionp. 23
Introductionp. 23
Exponential Distributionp. 23
Characterization of Exponential Distributionp. 27
Memoryless Propertyp. 27
Constant Failure Rate Functionp. 30
Extreme Value Distributionp. 30
Order Statistics and Exponential Distributionp. 32
Some Properties of Order Statisticsp. 32
Characterization Based on Order Statisticsp. 35
Record Valuesp. 37
More Applicationsp. 37
Problemsp. 40
Poisson Processp. 45
Poisson Process as a Counting Processp. 45
Characterization of Poisson Processes as Counting Processesp. 47
Poisson Process as a Renewal Processp. 53
Further Properties of Poisson Processp. 57
Superposition Processp. 57
Decomposition of Poisson Processp. 58
Examples of Poisson Processp. 60
Problemsp. 67
Parametric Families of Lifetime Distributionsp. 71
Weibull Distributionp. 71
Gamma Distributionp. 74
Change-Point Modelp. 78
Mixture Exponential Distributionp. 79
IFR (DFR) and Mixture Erlang Distributionp. 81
Problemsp. 84
Lifetime Distribution Classesp. 87
IFR and DFRp. 87
IFR and PF2p. 87
Smoothness of IFR Distributionp. 90
A Sufficient Conditionp. 91
IFRA and DFRA Classesp. 92
Several Lifetime Distribution Classesp. 95
Preservation of Lifetime Distributions Under Reliability Operationsp. 99
Independent Sumsp. 99
Mixture of Lifetime Distributionsp. 101
Shock Models and Lifetime Distribution Classesp. 104
IFRA Property of Shock Modelp. 104
Extension of Cumulative Damage Modelp. 107
General Cumulative Damage Modelp. 108
Shock Models Leading to Other Lifetime Distributionsp. 110
Problemsp. 112
Multivariate Lifetime Distributionsp. 117
Basic Properties of Bivariate Distributionsp. 117
Bivariate Memoryless Propertyp. 120
Property of the BVEp. 125
A Nonfatal Shock Modelp. 133
Absolutely Continuous Bivariate Exponential Extensionsp. 135
Problemsp. 139
Association and Dependencep. 141
Several Concepts of Associationp. 141
MTP2 Distributionp. 146
Multivariate Failure Rate and Distribution classp. 149
Negative Associationp. 151
Problemsp. 156
Renewal Theoryp. 159
Renewal Theoremp. 159
High-Order Approximations and Boundsp. 163
Delayed Renewal Processp. 166
Defective Renewal Processp. 169
Problemsp. 175
Risk Theoryp. 179
Classical Risk Modelp. 179
Approximation and Bounds for Ruin Probabilityp. 181
Deficit at Ruinp. 183
Large Claim Casep. 185
Bounds in terms of NWU (NBU) Distribution Classesp. 186
Subexponential Classesp. 190
Risk Sharing and Stop-Loss Reinsurancep. 193
Problemsp. 196
Asset Pricing Theoryp. 199
Utility, Risk, and Pricing Kernelp. 199
Utility and Riskp. 199
Asset Pricing Formula and Pricing Kernelp. 200
Models for Returnsp. 203
ß-Representationp. 203
Frontier Expressionp. 204
Log-Normal Modelp. 204
Examples of Risk Assetsp. 205
Risk-Neutral Probabiltiesp. 207
Option Pricing for Binomial Modelp. 208
Pricing Formula for Multiple Stagesp. 208
Binomial Modelp. 208
Portfolio Managementp. 210
Discrete Financial Marketp. 210
Risk Managementp. 211
Hedging Optionsp. 213
Black-Scholes Formulap. 216
Problemsp. 218
Credit Risk Modelingp. 221
Two Models for Default Probabilityp. 221
Basic Notationp. 221
Reduced Formp. 222
Structural Modelp. 224
Valuation of Default Riskp. 225
No Recovery Zero-Coupon Defaultable Bondp. 226
Non-Zero Recoveryp. 226
Actual and Risk Neutral Default Intensityp. 227
Credit Rating: Default and Transitionp. 227
Credit Ratingp. 227
Rating Assignmentp. 229
Rating Transitionp. 229
Correlated Defaultsp. 230
Credit Metricsp. 230
Correlated Default Intensitiesp. 231
Copula-Based Correlation Modelingp. 231
Credit Derivativesp. 232
Credit Default Swapsp. 233
Collateral Debt Obligationsp. 234
Problemsp. 235
Bibliographical Notes and Further Readingp. 237
Referencesp. 241
Answers and Solutions to Selected Problemsp. 245
Indexp. 265
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780817649869
ISBN-10: 0817649867
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 267
Published: 2nd September 2010
Country of Publication: US
Dimensions (cm): 23.39 x 15.6  x 1.75
Weight (kg): 0.57

Earn 232 Qantas Points
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