Igor Rychlik & Jesper Ryden. Probability and Risk Analysis: An
Introduction for Engineers, Springer, 2006
The purpose of this book is to present concepts in a statistical treatment of risks. Such knowledge facilitates the understanding of the influence of random phenomena and gives a deeper knowledge of the possibilities offered by and algorithms found in certain software packages. Since Bayesian methods are frequently used in this field, a reasonable proportion of the presentation is devoted to such techniques.
The text is written with student in mind – a student who has studied elementary undergraduate courses in engineering mathematics, may be including a minor course in statistics. Even though we use a style of presentation traditionally found in the math literature (including descriptions like definitions, examples, etc. ), emphasis is put on the understanding of the theory and methods presented; hence reasoning of an informal character is frequent. With respect to the contents (and its presentation), the idea has not been to write another textbook on elementary probability and statistics — there are plenty of such books — but to focus on applications within the field of risk and safety analysis.
Each chapter ends with a section on exercises; short solutions are given in appendix. Especially in the first chapters, some exercises merely check basic concepts introduced, with no clearly attached application indicated. However, among the collection of exercises as a whole, the ambition has been to present problems of an applied character and to a great extent real data sets have been used when constructing the problems.
Our ideas have been the following for the structuring of the chapters: In Chapter 1, we introduce probabilities of events, including notions like independence and conditional probabilities. Chapter 2 aims at presenting the two fundamental ways of interpreting probabilities: the frequentist and the Bayesian. The concept of intensity, important in risk calculations and referred to in later chapters, as well as the notion of a stream of events is also introduced here. A condensed summary of properties for random variables and characterisation of distributions is given in Chapter 3; in particular, typical distributions met in risk analysis are presented and exemplified here. In Chapter 4 the most important notions of classical inference (point estimation, confidence intervals) are discussed and we also provide a short introduction to bootstrap methodology. Further topics on probability are presented in Chapter 5, where notions like covariance, correlation, and conditional distributions are discussed.
The second part of the book, Chapters 6-10, are oriented at different types of problems and applications found in risk and safety analysis. In Chapter 6 we treat two problems: estimation of a probability for some (undesirable) event and estimation of the mean in a Poisson distribution (that is, the constant risk for accidents). The concept of conjugated priors to facilitate the computation of posterior distributions is introduced. Chapter 7 relates to notions introduced in Chapter 2 – intensities of events (accidents) and streams of events. By now the reader has hopefully reached a higher level of understanding and applying techniques from probability and statistics. Further topics can therefore be introduced, like lifetime analysis and Poisson regression. Discussion of absolute risks and tolerable risks is given. Furthermore, an orientation on more general Poisson processes (e.g. in the plane) is found. In structural engineering, safety indices are frequently used in design regulations. In Chapter 8, a discussion on such indices is given, as well as remarks on their computation. In this context, we discuss Gauss’ approximation formulae, which can be used to compute the values of indices approximately. More generally speaking, Gauss’ approximation formulae render approximations of the expected value and variance for functions of random variables. Moreover, approximate confidence intervals can be obtained in those situations by the so-called delta method, introduced at the end of the chapter. In Chapter 9, focus is on how to estimate characteristic values used in design codes and norms. First, a parametric approach is presented, thereafter an orientation on the POT (Peaks Over Threshold) method is given. Finally, in Chapter 10, an introduction to statistical extreme-value distributions is given. Much of the discussion is related to calculation of design loads and retu periods.
The purpose of this book is to present concepts in a statistical treatment of risks. Such knowledge facilitates the understanding of the influence of random phenomena and gives a deeper knowledge of the possibilities offered by and algorithms found in certain software packages. Since Bayesian methods are frequently used in this field, a reasonable proportion of the presentation is devoted to such techniques.
The text is written with student in mind – a student who has studied elementary undergraduate courses in engineering mathematics, may be including a minor course in statistics. Even though we use a style of presentation traditionally found in the math literature (including descriptions like definitions, examples, etc. ), emphasis is put on the understanding of the theory and methods presented; hence reasoning of an informal character is frequent. With respect to the contents (and its presentation), the idea has not been to write another textbook on elementary probability and statistics — there are plenty of such books — but to focus on applications within the field of risk and safety analysis.
Each chapter ends with a section on exercises; short solutions are given in appendix. Especially in the first chapters, some exercises merely check basic concepts introduced, with no clearly attached application indicated. However, among the collection of exercises as a whole, the ambition has been to present problems of an applied character and to a great extent real data sets have been used when constructing the problems.
Our ideas have been the following for the structuring of the chapters: In Chapter 1, we introduce probabilities of events, including notions like independence and conditional probabilities. Chapter 2 aims at presenting the two fundamental ways of interpreting probabilities: the frequentist and the Bayesian. The concept of intensity, important in risk calculations and referred to in later chapters, as well as the notion of a stream of events is also introduced here. A condensed summary of properties for random variables and characterisation of distributions is given in Chapter 3; in particular, typical distributions met in risk analysis are presented and exemplified here. In Chapter 4 the most important notions of classical inference (point estimation, confidence intervals) are discussed and we also provide a short introduction to bootstrap methodology. Further topics on probability are presented in Chapter 5, where notions like covariance, correlation, and conditional distributions are discussed.
The second part of the book, Chapters 6-10, are oriented at different types of problems and applications found in risk and safety analysis. In Chapter 6 we treat two problems: estimation of a probability for some (undesirable) event and estimation of the mean in a Poisson distribution (that is, the constant risk for accidents). The concept of conjugated priors to facilitate the computation of posterior distributions is introduced. Chapter 7 relates to notions introduced in Chapter 2 – intensities of events (accidents) and streams of events. By now the reader has hopefully reached a higher level of understanding and applying techniques from probability and statistics. Further topics can therefore be introduced, like lifetime analysis and Poisson regression. Discussion of absolute risks and tolerable risks is given. Furthermore, an orientation on more general Poisson processes (e.g. in the plane) is found. In structural engineering, safety indices are frequently used in design regulations. In Chapter 8, a discussion on such indices is given, as well as remarks on their computation. In this context, we discuss Gauss’ approximation formulae, which can be used to compute the values of indices approximately. More generally speaking, Gauss’ approximation formulae render approximations of the expected value and variance for functions of random variables. Moreover, approximate confidence intervals can be obtained in those situations by the so-called delta method, introduced at the end of the chapter. In Chapter 9, focus is on how to estimate characteristic values used in design codes and norms. First, a parametric approach is presented, thereafter an orientation on the POT (Peaks Over Threshold) method is given. Finally, in Chapter 10, an introduction to statistical extreme-value distributions is given. Much of the discussion is related to calculation of design loads and retu periods.