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Statistics and Probability Theory
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ENGINEERING DECISIONS UNDER UNCERTAINTY.- Lecture 1.- 1.1 Introduction.- 1.2 Societal Decision Making and Risk.- 1.2.1 Example 1.1 - Feasibility of Hydraulic Power Plant .- 1.3 Definition of Risk.- 1.4 Self Assessment Questions / Exercises .- 2 BASIC PROBABILITY THEORY .- Lecture 2.- 2.1 Introduction .- 2.2 Definition of Probability.- 2.2.1 Frequentistic Definition.- 2.2.3 Bayesian Definition.- 2.2.4 Practical Implications of the Different Interpretations of Probability.- 2.3 Sample Space and Events.- 2.4 The three Axioms of Probability Theory.- 2.5 Conditional Probability and Bayes' Rule.- 2.5.1 Example 2.1 - Using Bayes' Rule for Concrete Assessment .- 2.5.2 Example 2.2 - Using Bayes' Rule for Bridge Upgrading.- 2.6 Self Assessment Questions / Exercises.- 3 DESCRIPTIVE STATISTICS.- Lecture 3 .- 3.1 Introduction.- 3.2 Numerical Summaries.- 3.2.1 Central Measures.- 3.2.2 Example 3.1 - Concrete Compressive Strength Data.- 3.2.3 Example 3.2 - Traffic Flow Data.- 3.2.4 Dispersion Measures.- 3.2.5 Other Measures.- 3.2.6 Sample Moments and Sample Central Moments.- 3.2.7 Measures of Correlation.- 3.3 Graphical Representations.- 3.3.1 One-Dimensional Scatter Diagrams.- 3.3.2 Histograms.- 3.3.3 Quantile Plots.- 3.3.4 Tukey Box Plots.- 3.3.5 Q-Q Plots and Tukey Mean-Difference Plot.- 3.4 Self Assessment Questions / Exercises.- 4 UNCERTAINTY MODELLING.- Lecture 4.- 4.1 Introduction.- 4.2 Uncertainties in Engineering Problems.- 4.3 Random Variables.- 4.3.1 Cumulative Distribution and Probability Density Functions.- 4.3.2 Moments of Random Variables and the Expectation Operator.- 4.3.3 Example 4.1 - Uniform distribution.- Lecture 5.- 4.3.4 Properties of the Expectation Operator.- 4.3.5 Random Vectors and Joint Moments.- 4.3.6 Example 4.2 - linear combinations and random variables.- 4.3.7 Conditional Distributions and Conditional Moments .- 4.3.8 The Probability Distribution for the Sum of two Random Variables .- 4.3.9 Example 4.3 - Density Function for the Sum of two Random Variables - Special Case Normal Distribution.- 4.3.10 The Probability Distribution for Functions of Random Variables .- 4.3.11 Example 4.4 - Probability Distribution for a Function of Random Variables.- Lecture 6.- 4.3.12 Probability Density and Distribution Functions.- 4.3.13 The Central Limit Theorem and Derived Distributions.- 4.3.14 Example 4.5 - Central Limit Theorem.- 4.3.15 The Normal Distribution.- 4.3.16 The Lognormal Distribution.- 4.4 Stochastic Processes and Extremes.- 4.4.1 Random Sequences - Bernoulli Trials.- 4.4.2 Example 4.6 - Quality Control of Concrete.- Lecture 7 .- 4.4.3 The Poisson Counting Process .- 4.4.4 Continuous Random Processes.- 4.4.5 Stationarity and Ergodicity.- 4.4.6 Statistical Assessment of Extreme Values.- 4.4.7 Extreme Value Distributions.- 4.4.8 Type I Extreme Maximum Value Distribution - Gumbel max.- 4.4.9 Type I Extreme Minimum Value Distribution - Gumbel min.- 4.4.10 Type II Extreme Maximum Value Distribution - Frechet max.- 4.4.11 Type III Extreme Minimum Value Distribution - Weibull min.- 4.4.12 Return Period for Extreme Events.- 4.4.13 Example 4.7 - A Flood with a 100-Year Return Period.- 4.5 Self Assessment Questions / Exercises.- 5 ESTIMATION AND MODEL BUILDING.- Lecture 8.- 5.1 Introduction .- 5.2 Selection of Probability Distributions.- 5.2.1 Model Selection by Use of Probability Paper.- 5.3 Estimation of Distribution Parameters.- 5.3.1 The Method of Moments.- 5.3.2 The Method of Maximum Likelihood.- 5.3.3 Example 5.1 - Parameter Estimation.- Lecture 9.- 5.4 Bayesian Estimation Methods.- 5.4.1 Example 5.2 - Yield Stress of a Steel Bar.- 5.5 Bayesian Regression Analysis.- 5.5.1 Linear Regression: Prior Model.- 5.5.2 Example 5.3 - Tensile Strength of Timber: Prior Model.- 5.5.3 Updating Regression Coefficients: Posterior Model.- 5.5.4 Example 5.4 - Updating Regression Coefficients (determined in Example 5.3).- Lecture 10.- 5.6 Probability Distributions in Statistics.- 5.6.1 The Chi-Square (c2)-Distribution.- 5.6.2 The Chi (c)-Distribution.- 5.7 Estimators for Sample Descriptors - Sample Statistics.- 5.7.1 Statistical Characteristics of the Sample Average .- 5.7.2 Statistical Characteristics of the Sample Variance.- 5.7.3 Confidence Intervals.- 5.8 Testing for Statistical Significance.- 5.8.1 The Hypothesis Testing Procedure .- 5.8.2 Testing of the Mean with Known Variance.- 5.8.3 Some Remarks on Testing.- Lecture 11.- 5.9 Model Evaluation by Statistical Testing.- 5.9.1 The Chi-Square (c2)-Goodness of Fit Test.- 5.9.2 The Kolmogorov-Smirnov Goodness of Fit Test.- 5.9.3 Model Comparison.- 5.10 Self Assessment Questions / Exercises.- 6 METHODS OF STRUCTURAL RELIABILITY.- Lecture 12.- 6.1 Introduction.- 6.2 Failure Events and Basic Random Variables.- 6.3 Linear Limit State Functions and Normal Distributed Variables.- 6.3.1 Example 6.1 - Reliability of a Steel Rod - Linear Safety Margin.- 6.4 The Error Propagation Law.- 6.4.1 Example 6.2 - Error Propagation Law.- 6.5 Non-linear Limit State Functions.- 6.5.1 Example 6.3 - FORM - Non-linear Limit State Function.- 6.6 Simulation Methods.- 6.6.1 Example 6.4: Monte Carlo Simulation.- 6.7 Self Assessment Questions / Exercises.- 7 BAYESIAN DECISION ANALYSIS.- Lecture 13.- 7.1 Introduction.- 7.2 The Decision / Event Tree.- 7.3 Decisions Based on Expected Values.- 7.4 Decision Making Subject to Uncertainty.- 7.5 Decision Analysis with Given Information - Prior Analysis.- 7.6 Decision Analysis with Additional Information - Posterior Analysis.- 7.7 Decision Analysis with 'Unknown' Information - Pre-posterior Analysis.- 7.8 The Risk Treatment Decision Problem.- 7.9 Self Assessment Questions / Exercises.- A ANSWERS TO SELF ASSESSMENT QUESTIONS.- A.1 Chapter 1.- A.2 Chapter 2.- A.3 Chapter 3.- A.4 Chapter 4.- A.5 Chapter 5.- A.6 Chapter 6.- A.7 Chapter 7.- B EXAMPLES OF CALCULATIONS.- B.1 Chapter 5.- B.1.1 Equation 5.67.- B.1.2 Equation 5.71.- B.1.3 Examples on Chi-square significance test.- B.2 Chapter 6.- B.2.1 Example 6.2.- B.2.2 Example 6.3.- C TABLES.- References.- Index .

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