Probability and risk analysis : an introduction for engineers

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Title:

Personal Author:

Rychlik, Igor

Publication Information:

Berlin : Springer, 2006

ISBN:

9783540242239

Subject Term:

Engineering mathematics

Probabilities

Reliability (Engineering)

Added Author:

Ryden, Jesper

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010132930	TA330 R92 2006	Open Access Book	Book	Searching... Unknown
Searching... PSZ JB	30000010087597	TA330 R92 2006	Open Access Book	Book	Searching... Unknown

The purpose of this book is to present concepts in a statistical treatment of risks. Such knowledge facilitates the understanding of the in?uence of random phenomena and gives a deeper knowledge of the possibilities o?ered by and algorithms found in certain software packages. Since Bayesian methods are frequently used in this ?eld, a reasonable proportion of the presentation is devoted to such techniques. The text is written with student in mind - a student who has studied - ementary undergraduate courses in engineering mathematics, may be incl- ing a minor course in statistics. Even though we use a style of presentation traditionally found in the math literature (including descriptions like de- 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 ?eld of risk and safety analysis. Each chapter ends with a section on exercises; short solutions are given in appendix. Especially in the ?rst 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.

1 Basic Probability	p. 1
1.1 Sample Space, Events, and Probabilities	p. 4
1.2 Independence	p. 8
1.2.1 Counting variables	p. 10
1.3 Conditional Probabilities and the Law of Total Probability	p. 12
1.4 Event-tree Analysis	p. 15
2 Probabilities in Risk Analysis	p. 21
2.1 Bayes' Formula	p. 22
2.2 Odds and Subjective Probabilities	p. 23
2.3 Recursive Updating of Odds	p. 27
2.4 Probabilities as Long-term Frequencies	p. 30
2.5 Streams of Events	p. 33
2.6 Intensities of Streams	p. 37
2.6.1 Poisson streams of events	p. 40
2.6.2 Non-stationary streams	p. 43
3 Distributions and Random Variables	p. 49
3.1 Random Numbers	p. 51
3.1.1 Uniformly distributed random numbers	p. 51
3.1.2 Non-uniformly distributed random numbers	p. 52
3.1.3 Examples of random numbers	p. 54
3.2 Some Properties of Distribution Functions	p. 55
3.3 Scale and Location Parameters - Standard Distributions	p. 59
3.3.1 Some classes of distributions	p. 60
3.4 Independent Random Variables	p. 62
3.5 Averages - Law of Large Numbers	p. 63
3.5.1 Expectations of functions of random variables	p. 65
4 Fitting Distributions to Data - Classical Inference	p. 69
4.1 Estimates of F[subscript x]	p. 72
4.2 Choosing a Model for F[subscript x]	p. 74
4.2.1 A graphical method: probability paper	p. 75
4.2.2 Introduction to [chi superscript 2]-method for goodness-of-fit tests	p. 77
4.3 Maximum Likelihood Estimates	p. 80
4.3.1 Introductory example	p. 80
4.3.2 Derivation of ML estimates for some common models	p. 82
4.4 Analysis of Estimation Error	p. 85
4.4.1 Mean and variance of the estimation error [epsilon]	p. 86
4.4.2 Distribution of error, large number of observations	p. 89
4.5 Confidence Intervals	p. 92
4.5.1 Introduction. Calculation of bounds	p. 92
4.5.2 Asymptotic intervals	p. 94
4.5.3 Bootstrap confidence intervals	p. 95
4.5.4 Examples	p. 95
4.6 Uncertainties of Quantiles	p. 98
4.6.1 Asymptotic normality	p. 98
4.6.2 Statistical bootstrap	p. 100
5 Conditional Distributions with Applications	p. 105
5.1 Dependent Observations	p. 105
5.2 Some Properties of Two-dimensional Distributions	p. 107
5.2.1 Covariance and correlation	p. 113
5.3 Conditional Distributions and Densities	p. 115
5.3.1 Discrete random variables	p. 115
5.3.2 Continuous random variables	p. 116
5.4 Application of Conditional Probabilities	p. 117
5.4.1 Law of total probability	p. 117
5.4.2 Bayes' formula	p. 118
5.4.3 Example: Reliability of a system	p. 119
6 Introduction to Bayesian Inference	p. 125
6.1 Introductory Examples	p. 126
6.2 Compromising Between Data and Prior Knowledge	p. 130
6.2.1 Bayesian credibility intervals	p. 132
6.3 Bayesian Inference	p. 132
6.3.1 Choice of a model for the data - conditional independence	p. 133
6.3.2 Bayesian updating and likelihood functions	p. 134
6.4 Conjugated Priors	p. 135
6.4.1 Unknown probability	p. 137
6.4.2 Probabilities for multiple scenarios	p. 139
6.4.3 Priors for intensity of a stream A	p. 141
6.5 Remarks on Choice of Priors	p. 143
6.5.1 Nothing is known about the parameter [theta]	p. 143
6.5.2 Moments of [Theta] are known	p. 144
6.6 Large number of observations: Likelihood dominates prior density	p. 147
6.7 Predicting Frequency of Rare Accidents	p. 151
7 Intensities and Poisson Models	p. 157
7.1 Time to the First Accident - Failure Intensity	p. 157
7.1.1 Failure intensity	p. 157
7.1.2 Estimation procedures	p. 162
7.2 Absolute Risks	p. 166
7.3 Poisson Models for Counts	p. 170
7.3.1 Test for Poisson distribution - constant mean	p. 171
7.3.2 Test for constant mean - Poisson variables	p. 173
7.3.3 Formulation of Poisson regression model	p. 174
7.3.4 ML estimates of [Beta subscript 0],...,[Beta subscript p]	p. 180
7.4 The Poisson Point process	p. 182
7.5 More General Poisson Processes	p. 185
7.6 Decomposition and Superposition of Poisson Processes	p. 187
8 Failure Probabilities and Safety Indexes	p. 193
8.1 Functions Often Met in Applications	p. 194
8.1.1 Linear function	p. 194
8.1.2 Often used non-linear function	p. 198
8.1.3 Minimum of variables	p. 201
8.2 Safety Index	p. 202
8.2.1 Cornell's index	p. 202
8.2.2 Hasofer-Lind index	p. 204
8.2.3 Use of safety indexes in risk analysis	p. 204
8.2.4 Return periods and safety index	p. 205
8.2.5 Computation of Cornell's index	p. 206
8.3 Gauss' Approximations	p. 207
8.3.1 The delta method	p. 209
9 Estimation of Quantiles	p. 217
9.1 Analysis of Characteristic Strength	p. 217
9.1.1 Parametric modelling	p. 218
9.2 The Peaks Over Threshold (POT) Method	p. 220
9.2.1 The POT method and estimation of [kappav][alpha] quantiles	p. 222
9.2.2 Example: Strength of glass fibres	p. 223
9.2.3 Example: Accidents in mines	p. 224
9.3 Quality of Components	p. 226
9.3.1 Binomial distribution	p. 227
9.3.2 Bayesian approach	p. 228
10 Design Loads and Extreme Values	p. 231
10.1 Safety Factors, Design Loads, Characteristic Strength	p. 232
10.2 Extreme Values	p. 233
10.2.1 Extreme-value distributions	p. 234
10.2.2 Fitting a model to data: An example	p. 240
10.3 Finding the 100-year Load: Method of Yearly Maxima	p. 241
10.3.1 Uncertainty analysis of S[subscript T]: Gumbel case	p. 242
10.3.2 Uncertainty analysis of S[subscript T]: GEV case	p. 244
10.3.3 Warning example of model error	p. 245
10.3.4 Discussion on uncertainty in design-load estimates	p. 247
A Some Useful Tables	p. 251
Short Solutions to Problems	p. 257
References	p. 275
Index	p. 279

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Table of Contents