Faber Statistics and Probability Theory

In Pursuit of Engineering Decision Support
1. Auflage 2012
ISBN: 978-94-007-4056-3
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 192 Seiten

Reihe: Engineering (R0)

This book provides the reader with the basic skills and tools
of statistics and probability in the context of engineering modeling and analysis. The emphasis is on the application and the reasoning behind the application of these skills and tools for the purpose of enhancing  decision making in engineering.

The purpose of the book is to ensure that the reader will acquire the required theoretical basis and technical skills such as to feel comfortable with the theory of basic statistics and probability. Moreover, in this book, as opposed to many standard books on the same subject, the perspective is to focus on the use of the theory for the purpose of engineering model building and decision making.  This work is suitable for readers with little or no prior knowledge on the subject of statistics and probability.

Autoren/Hrsg.

Faber, Michael Havbro

Inhaltsverzeichnis

1;Statistics and Probability Theory;4
1.1;Preface;7
1.2;Acknowledgements;10
1.3;Contents;11
2;Chapter 1: Engineering Decisions Under Uncertainty;15
2.1;Lecture 1 (Aim of the Present Lecture);15
2.2;1.1 Introduction;15
2.3;1.2 Societal Decision Making and Risk;16
2.3.1;1.2.1 Example 1.1-Feasibility of Hydraulic Power Plant;17
2.4;1.3 De?nition of Risk;20
2.5;1.4 Self Assessment Questions/Exercises;21
3;Chapter 2: Basic Probability Theory;22
3.1;Lecture 2 (Aim of the Present Lecture);22
3.2;2.1 Introduction;22
3.3;2.2 De?nition of Probability;23
3.3.1;2.2.1 Frequentistic De?nition;23
3.3.2;2.2.2 Classical De?nition;24
3.3.3;2.2.3 Bayesian De?nition;24
3.3.4;2.2.4 Practical Implications of the Different Interpretations of Probability;25
3.4;2.3 Sample Space and Events;25
3.5;2.4 The Three Axioms of Probability Theory;27
3.6;2.5 Conditional Probability and Bayes' Rule;27
3.6.1;2.5.1 Example 2.1-Using Bayes' Rule for Concrete Assessment;29
3.6.2;2.5.2 Example 2.2-Using Bayes' Rule for Bridge Upgrading;30
3.7;2.6 Self Assessment Questions/Exercises;31
4;Chapter 3: Descriptive Statistics;34
4.1;Lecture 3 (Aim of the Present Lecture);34
4.2;3.1 Introduction;34
4.3;3.2 Numerical Summaries;35
4.3.1;3.2.1 Central Measures;35
4.3.2;3.2.2 Example 3.1-Concrete Compressive Strength Data;35
4.3.3;3.2.3 Example 3.2-Traf?c Flow Data;36
4.3.4;3.2.4 Dispersion Measures;37
4.3.5;3.2.5 Other Measures;38
4.3.6;3.2.6 Sample Moments and Sample Central Moments;39
4.3.7;3.2.7 Measures of Correlation;39
4.4;3.3 Graphical Representations;40
4.4.1;3.3.1 One-Dimensional Scatter Diagrams;41
4.4.2;3.3.2 Histograms;41
4.4.3;3.3.3 Quantile Plots;44
4.4.4;3.3.4 Tukey Box Plots;49
4.4.5;3.3.5 Q-Q Plots and Tukey Mean-Difference Plot;52
4.5;3.4 Self Assessment Questions/Exercises;54
5;Chapter 4: Uncertainty Modeling;56
5.1;Lecture 4 (Aim of the Present Lecture);56
5.2;4.1 Introduction;56
5.3;4.2 Uncertainties in Engineering Problems;57
5.4;4.3 Random Variables;59
5.4.1;4.3.1 Cumulative Distribution and Probability Density Functions;60
5.4.2;4.3.2 Moments of Random Variables and the Expectation Operator;61
5.4.3;4.3.3 Example 4.1-Uniform Distribution;62
5.4.3.1;Lecture 5 (Aim of the Present Lecture);63
5.4.4;4.3.4 Properties of the Expectation Operator;64
5.4.5;4.3.5 Random Vectors and Joint Moments;65
5.4.6;4.3.6 Example 4.2-Linear Combinations and Random Variables;66
5.4.7;4.3.7 Conditional Distributions and Conditional Moments;67
5.4.8;4.3.8 The Probability Distribution for the Sum of Two Random Variables;68
5.4.9;4.3.9 Example 4.3-Density Function for the Sum of Two Random Variables-Special Case Normal Distribution;70
5.4.10;4.3.10 The Probability Distribution for Functions of Random Variables;71
5.4.11;4.3.11 Example 4.4-Probability Distribution for a Function of Random Variables;72
5.4.11.1;Lecture 6 (Aim of the Present Lecture);74
5.4.12;4.3.12 Probability Density and Distribution Functions;74
5.4.13;4.3.13 The Central Limit Theorem and Derived Distributions;75
5.4.14;4.3.14 Example 4.5-Central Limit Theorem;77
5.4.15;4.3.15 The Normal Distribution;77
5.4.16;4.3.16 The Lognormal Distribution;80
5.5;4.4 Stochastic Processes and Extremes;80
5.5.1;4.4.1 Random Sequences-Bernoulli Trials;81
5.5.2;4.4.2 Example 4.6-Quality Control of Concrete;82
5.5.2.1;Lecture 7 (Aim of the Present Lecture);82
5.5.3;4.4.3 The Poisson Counting Process;83
5.5.4;4.4.4 Continuous Random Processes;84
5.5.5;4.4.5 Stationarity and Ergodicity;87
5.5.6;4.4.6 Statistical Assessment of Extreme Values;88
5.5.7;4.4.7 Extreme Value Distributions;89
5.5.8;4.4.8 Type I Extreme Maximum Value Distribution-Gumbel Max;90
5.5.9;4.4.9 Type I Extreme Minimum Value Distribution-Gumbel Min;92
5.5.10;4.4.10 Type II Extreme Maximum Value Distribution-Fréchet Max;92
5.5.11;4.4.11 Type III Extreme Minimum Value Distribution-Weibull Min;93
5.5.12;4.4.12 Return Period for Extreme Events;94
5.5.13;4.4.13 Example 4.7-A Flood with a 100-Year Return Period;94
5.6;4.5 Self Assessment Questions/Exercises;95
6;Chapter 5: Estimation and Model Building;98
6.1;Lecture 8 (Aim of the Present Lecture);98
6.2;5.1 Introduction;99
6.3;5.2 Selection of Probability Distributions;100
6.3.1;5.2.1 Model Selection by Use of Probability Paper;101
6.4;5.3 Estimation of Distribution Parameters;104
6.4.1;5.3.1 The Method of Moments;104
6.4.2;5.3.2 The Method of Maximum Likelihood;105
6.4.3;5.3.3 Example 5.1-Parameter Estimation;106
6.4.3.1;Lecture 9 (Aim of the Present Lecture);108
6.5;5.4 Bayesian Estimation Methods;109
6.5.1;5.4.1 Example 5.2-Yield Stress of a Steel Bar;110
6.6;5.5 Bayesian Regression Analysis;112
6.6.1;5.5.1 Linear Regression: Prior Model;113
6.6.2;5.5.2 Example 5.3-Tensile Strength of Timber: Prior Model;115
6.6.3;5.5.3 Updating Regression Coef?cients: Posterior Model;117
6.6.4;5.5.4 Example 5.4-Updating Regression Coef?cients (Determined in Example 5.3);117
6.6.4.1;Lecture 10 (Aim of the Present Lecture);119
6.7;5.6 Probability Distributions in Statistics;119
6.7.1;5.6.1 The Chi-Square (chi2)-Distribution;120
6.7.2;5.6.2 The Chi (chi)-Distribution;121
6.8;5.7 Estimators for Sample Descriptors-Sample Statistics;121
6.8.1;5.7.1 Statistical Characteristics of the Sample Average;122
6.8.2;5.7.2 Statistical Characteristics of the Sample Variance;124
6.8.3;5.7.3 Con?dence Intervals;125
6.9;5.8 Testing for Statistical Signi?cance;126
6.9.1;5.8.1 The Hypothesis Testing Procedure;127
6.9.2;5.8.2 Testing of the Mean with Known Variance;129
6.9.3;5.8.3 Some Remarks on Testing;129
6.9.3.1;Lecture 11 (Aim of the Present Lecture);130
6.10;5.9 Model Evaluation by Statistical Testing;130
6.10.1;5.9.1 The Chi-Square (chi2)-Goodness of Fit Test;131
6.10.2;5.9.2 The Kolmogorov-Smirnov Goodness of Fit Test;135
6.10.3;5.9.3 Model Comparison;137
6.11;5.10 Self Assessment Questions/Exercises;138
7;Chapter 6: Methods of Structural Reliability;141
7.1;Lecture 12 (Aim of the Present Lecture);141
7.2;6.1 Introduction;141
7.3;6.2 Failure Events and Basic Random Variables;142
7.4;6.3 Linear Limit State Functions and Normal Distributed Variables;143
7.4.1;6.3.1 Example 6.1-Reliability of a Steel Rod-Linear Safety Margin;144
7.5;6.4 The Error Propagation Law;145
7.5.1;6.4.1 Example 6.2-Error Propagation Law;147
7.6;6.5 Non-linear Limit State Functions;148
7.6.1;6.5.1 Example 6.3-FORM-Non-linear Limit State Function;149
7.7;6.6 Simulation Methods;151
7.7.1;6.6.1 Example 6.4: Monte Carlo Simulation;152
7.8;6.7 Self Assessment Questions/Exercises;154
8;Chapter 7: Bayesian Decision Analysis;155
8.1;Lecture 13 (Aim of the Present Lecture);155
8.2;7.1 Introduction;156
8.3;7.2 The Decision/Event Tree;156
8.4;7.3 Decisions Based on Expected Values;157
8.5;7.4 Decision Making Subject to Uncertainty;159
8.6;7.5 Decision Analysis with Given Information-Prior Analysis;159
8.7;7.6 Decision Analysis with Additional Information-Posterior Analysis;160
8.8;7.7 Decision Analysis with `Unknown' Information-Pre-posterior Analysis;163
8.9;7.8 The Risk Treatment Decision Problem;164
8.10;7.9 Self Assessment Questions/Exercises;166
9;Appendix A: Answers to Self Assessment Questions;167
9.1;A.1 Chapter 1;167
9.2;A.2 Chapter 2;168
9.3;A.3 Chapter 3;170
9.4;A.4 Chapter 4;171
9.5;A.5 Chapter 5;173
9.6;A.6 Chapter 6;175
9.7;A.7 Chapter 7;179
10;Appendix B: Examples of Calculations;182
10.1;B.1 Chapter 5;182
10.1.1;B.1.1 Equation 5.67;182
10.1.2;B.1.2 Equation 5.71;182
10.1.3;B.1.3 Examples on Chi-Square Signi?cance Test;183
10.2;B.2 Chapter 6;184
10.2.1;B.2.1 Example 6.2;184
10.2.2;B.2.2 Example 6.3;184
11;Appendix C: Tables;187
12;References;194
13;Index;195