Rychlik - Probability and Risk Analysis - An Introduction for Engineers (Springer, 2006).pdf

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Probability and Risk Analysis
Igor Rychlik Jesper Rydén
Probability and Risk Analysis
An Introduction for Engineers
With 46 Figures and 7 Tables
123
Dr.
Jesper Rydén
School of Technology and Society, Malmö University,
Prof. Igor Rychlik
Dept. of Mathematical Statistics,
Ö
Varvsg
11A,
Lund University,
SE-20506 Malmö, Sweden
Box 118,
22100 Lund, Sweden
e-mail : jesper.ryden@ts.mah.se
e-mail : igor@maths.lth.se
LibraryofCongressControlNumber:
2006925439
ISBN-10 3-540-24223-6 Springer Berlin Heidelberg New York
ISBN-13 978-3-540-24223-9 Springer Berlin Heidelberg New York
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Preface
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 el-
ementary undergraduate courses in engineering mathematics, may be includ-
ing a minor course in statistics. Even though we use a style of presentation
traditionally found in the math literature (including descriptions like defin-
itions, 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 indepen-
dence and conditional probabilities. Chapter 2 aims at presenting the two fun-
damental 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 im-
portant notions of classical inference (point estimation, confidence intervals)
VI
Preface
are discussed and we also provide a short introduction to bootstrap method-
ology. 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. Bayesian
methods are further discussed in Chapter 6. There 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 regu-
lations. In Chapter 8, a discussion on such indices is given, as well as remarks
on their computation. In this context, we discuss Gauss’ approximation formu-
lae, 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
return periods.
We are grateful to many students whose comments have improved the
presentation. Georg Lindgren has read the whole manuscript and given
many fruitful comments. Thanks also to Anders Bengtsson, Oskar Hagberg,
Krzysztof Nowicki, Niels C. Overgaard, and Krzysztof Podgórski for reading
parts of the manuscript; Tord Isaksson and Colin McIntyre for valuable re-
marks; and Tord Rikte and Klas Bogsjö for assistance with exercises. The
first author would like to express his gratitude to Jeanne Wéry for her long-
term encouragement and interest in his work. Finally, a special thanks to our
families for constant support and patience.
Lund and Malmö,
Igor Rychlik
March, 2006
Jesper Rydén

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