## Table of contents for Probability and risk analysis : an introduction for engineers / Igor Rychlik, Jesper Rydén.

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```1   Basic Probability ........................................ 1
1.1 Sample Space, Events, and Probabilities ..... . ........... 4
1.2 Independence ...........................   ............. 8
1.2.1 Counting variables............................. .. . 10
1.3 Conditional Probabilities and the Law of Total Probability ... 12
1.4  Event-tree Analysis ......................................  15
2   Probabilities in  Risk  Analysis ..............................  21
2.1 Bayes' Formula ........ ............   ................ 22
2.2 Odds and Subjective Probabilities ........................ 23
2.3 Recursive Updating of Odds .............................. 27
2.4 Probabilities as Long-term Frequencies ..................... 30
2.5 Streams of Events ............ . .  ........... ........ 33
2.6  Intensities of Streams ................  ...................  37
2.6.1 Poisson streams of events........................ . 40
2.6.2  Non-stationary streams  ................... ...... .  43
3   Distributions and Random Variables........... ...      ....... 49
3.1 Random Numbers ................................... .. 51
3.1.1 Uniformly distributed random numbers .............. 51
3.1.2 Non-uniformly distributed random numbers........... 52
3.1.3 Examples of random numbers ....................  . . 54
3.2 Some Properties of Distribution Functions .................. 55
3.3 Scale and Location Parameters - Standard Distributions ..... 59
3.3.1 Some classes of distributions ..................... 60
3.4 Independent Random Variables .......................... 62
3.5 Averages - Law of Large Numbers ......................... 63
3.5.1 Expectations of functions of random variables ......... 65
4   Fitting Distributions to Data - Classical Inference ......... 69
4.1 Estimates of Fx ................ ................... .. 72
4.2 Choosing a Model for Fx ............................. 74
4.2.1 A graphical method: probability paper ............... 75
4.2.2 Introduction to X2-method for goodness-of-fit tests .... 77
4.3 Maximum Likelihood Estimates .......................... 80
4.3.1 Introductory example ............................. 80
4.3.2 Derivation of ML estimates for some common models .. 82
4.4  Analysis of Estimation  Error ..............................  85
4.4.1 Mean and variance of the estimation error E .......... 86
4.4.2 Distribution of error, large number of observations..... 89
4.5 Confidence Intervals ................ .................. 92
4.5.1 Introduction. Calculation of bounds ................. 92
4.5.2  Asymptotic  intervals ...............................  94
4.5.3 Bootstrap confidence intervals ................. . . .. 95
4.5.4 Examples ....................... .... ......... 95
4.6  Uncertainties of Quantiles  .... .................. .............  98
4.6.1  Asymptotic normality..............................  98
4.6.2  Statistical bootstrap  ............................ . 100
5   Conditional Distributions with Applications .............. 105
5.1 Dependent Observations.................. ...............105
5.2 Some Properties of Two-dimensional Distributions ............ 107
5.2.1 Covariance and correlation ......................... 113
5.3 Conditional Distributions and Densities .................... 115
5.3.1 Discrete random variables .......................... 115
5.3.2  Continuous random  variables  ....................... 116
5.4 Application of Conditional Probabilities .... ................ 117
5.4.1  Law  of total probability  ............... ............ 117
5.4.2 Bayes' formula ................................... 118
5.4.3  Example: Reliability of a system  .................... 119
6   Introduction to Bayesian Inference ........................ 125
6.1 Introductory Examples ................................. 126
6.2 Compromising Between Data and Prior Knowledge .......... 130
6.2.1  Bayesian credibility intervals ...... . ... ... . .... .. . . 132
6.3 Bayesian Inference.......... .. ....................... ...132
6.3.1 Choice of a model for the data - conditional
independence  ..................................... 133
6.3.2 Bayesian updating and likelihood functions .......... 134
6.4 Conjugated Priors ................. ..................  135
6.4.1 Unknown probability ........................... ..137
6.4.2 Probabilities for multiple scenarios .................. 139
6.4.3  Priors for intensity of a stream  A  ................... 141
6.5 Remarks on Choice of Priors .......................... 143
6.5.1 Nothing is known about the parameter 0 ............. 143
6.5,2 Moments of 0 are known ......................... 144
6.6 Large number of observations: Likelihood dominates prior
density... .......................................  ......147
6.7 Predicting Frequency of Rare Accidents .................... 151
7   Intensities and Poisson Models ..............................157
7.1 Time to the First Accident - Failure Intensity....... ...... 157
7.1,1 Failure intensity... .................  ......... 157
7.1.2  Estimation  procedures  ............................  162
7.2  Absolute Risks .................... ...................... 166
7.3 Poisson Models for Counts ............................. 170
7.3.1 Test for Poisson distribution - constant mean ......... 171
7.3.2 Test for constant mean - Poisson variables. ........... 173
7.3.3 Formulation of Poisson regression model............. . 174
7.3.4  ML  estimates of  o, ... , 13p  ......................... 180
7.4 The Poisson Point process ............... ............. 182
7.5 More General Poisson Processes ................... ...... . 185
7.6 Decomposition and Superposition of Poisson Processes ....... 187
8   Failure Probabilities and Safety Indexes ................... 193
8.1  Functions Often Met in Applications ....................... 194
8.1.1  Linear  function  ...............  .................  194
8.1.2 Often used non-linear function ..................... 198
8.1.3  Minimum  of variables ........................... . 201
8.2  Safety  Index  ................... ...............  .... 202
8.2.1  Cornell's  index  ..................................... 202
8.2.2 Hasofer-Lind index ..............................204
8.2.3 Use of safety indexes in risk analysis ................. 204
8.2.4 Return periods and safety index ..................... 205
8.2.5  Computation of Cornell's index  ..................... 206
8.3  Gauss' Approximations .................................. 207
8.3.1  The  delta  method  ...............................  . 209
9   Estimation  of Quantiles ................................. 217
9.1 Analysis of Characteristic Strength ........................217
9.1.1  Parametric  modelling  ............................. . 218
9.2 The Peaks Over Threshold (POT) Method ................. 220
9.2.1 The POT method and estimation of Xa quantiles ..... 222
9.2.2  Example: Strength of glass fibres .................... 223
9.2.3 Example: Accidents in mines ................. .. . . .  224
9.3 Quality of Components................ ............... 226
9.3.1  Binomial distribution  ........................... . . 227
9.3.2 Bayesian approach................  ............. 228
10 Design Loads and Extreme Values .........................231
10.1 Safety Factors, Design Loads, Characteristic Strength ........ 232
10.2  Extreme Values  ........................................233
10.2.1  Extreme-value distributions......................... 234
10.2.2 Fitting a model to data: An example ................. 240
10.3 Finding the 100-year Load: Method of Yearly Maxima ....... 241
10.3.1 Uncertainty analysis of ST: Gumbel case ............. 242
10.3.2 Uncertainty analysis of sT: GEV case. .............. 244
10.3.3 Warning example of model error .................... 245
10.3.4 Discussion on uncertainty in design-load estimates ..... 247
A   Some Useful Tables .................. ......................251
Short Solutions to Problems ................................257
References ...............................................     275

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Library of Congress subject headings for this publication: Engineering mathematics, Probabilities, Reliability (Engineering)