Probability and Statistical Models

Probability models are now a vital componentof every scienti c investigation. This book is intended to introduce basic ideas in stochastic modeling, with emphasis on models and techniques. These models lead to well-known parametric lifetime distributions, such as exponential, Weibull, and gamma distributions, as well as the change-point and mixture models. They also motivate us to consider more general notions of nonparametric lifetime distribution classes. Particular attention has been paid to their applications in reliability, insurance mathematics, and economics. The following topics are the focus in this volume: 1. Exponential Distributions and the Poisson Process; 2. Parametric Lifetime Distributions; 3. Nonparametric Lifetime Distribution Classes; 4. Multivariate Exponential Extensions; 5. Association and Dependence; 6. Renewal Theory; 7. Applications to Reliability, Insurance, Finance, and Credit Risk. Chapter1providesnotationandbasicresultsinprobabilitytheorythatareneeded in the consequent chapters. Chapters 2 and 3 are devoted to models related to exponential distribution and Poisson processes. Particular attentions is paid to the characterizations of exponential distribution and the Poisson process. Two of the most important properties that characterize exponential distribution: the lack of memory property and constant failure rate are discussed in detail. Then the g- eralizations of exponential distribution are examined in three directions: through its parametric form that leads to parametric families of lifetime distributions; via notionsof aging(such as monotonefailure rate) that lead to a varietyof lifetime d- tribution classes; and through lifetime distributions of multiple component systems that lead to multivariate (mainly bivariate) exponential extension.