Probability, Staüistics,

and Reliability

for Engineers and

Scientists Second Edition

Bilal M. Ayyub Richard H. McCuen

CHAPMAN & HALL/CRC A CRC Press Company

Boca Raton London New York Washington, D.C.

Table of Contents Chapter 1 Introduction 1 1.1. Introduction 1 1.2. Types of Uncertainty 5 1.3. Introduction to Simulation 9 1.4. Problems 17 1.5. Simulation Projects 19

Chapter 2 Data Description and Treatment 25 2.1. Introduction 26 2.2. Classification of Data 26 2.3. Graphical Description of Data 28 2.4. Histograms and Frequency Diagrams 36 2.5. Descriptive Measures 39 2.6. Applications 46 2.7. Analysis of Simulated Data 49 2.8. Problems 54 2.9. Simulation Projects 60

Chapter 3 Fundamentals of Probability 63 3.1. Introduction 64 3.2. Sample Spaces, Sets, and Events 64 3.3. Mathematics of Probability 69 3.4. Random Variables and their Probability Distributions 83 3.5. Moments 91 3.6. Application: Water Supply and Quality 101 3.7. Simulation and Probability Distributions 102 3.8. Problems 104 3.9. Simulation Projects 109

Chapter 4 Probability Distributions for Discrete Random Variables 111 4.1. Introduction 111 4.2. Bernoulli Distribution 112 4.3. Binomial Distribution 113 4.4. Geometrie Distribution 115 4.5. Poisson Distribution 116 4.6. Negative Binomial and Pascal Probability Distributions 118 4.7. Hypergeometric Probability Distribution 118 4.8. Applications 119 4.9. Simulation of Discrete Random Variables 121 4.10. Problems 127 4.11. Simulation Projects 129

Chapter 5 Probability Distributions for Continuous Random Variables 131 5.1. Introduction 132 5.2. Uniform Distribution 132 5.3. Normal Distribution 134 5.4. Lognormal Distribution 138 5.5. Exponential Distribution 141 5.6. Triangulär Distribution 143 5.7. Gamma Distribution 144 5.8. Rayleigh Distribution 145 5.9. Statistical Probability Distributions 146 5.10. Extreme Value Distributions 149 5.11. Applications 155 5.12. Simulation and Probability Distributions 157 5.13. Problems 160 5.14. Simulation Projects 161

Chapter 6 Multiple Random Variables 165 6.1. Introduction 165 6.2. Joint Random Variables and their Probability Distributions 166 6.3. Functions of Random Variables 182 6.4. Applications 192 6.5. Multivariable Simulation 199 6.6. Problems 209 6.7. Simulation Projects 213

Chapter 7 Simulation 215 7.1. Introduction 216 7.2. Monte Carlo Simulation 221 7.3. Random Numbers 222 7.4. Generation of Random Variables 225 7.5. Generation of Selected Discrete Random Variables 232 7.6. Generation of Selected Continuous Random Variables 238 7.7. Applications 242 7.8. Problems 252 7.9. Simulation Projects 257

Chapter 8 Fundamentals of Statistical Analysis 259 8.1. Introduction 259 8.2. Estimation of Parameters 261 8.3. Sampling Distributions 276 8.4. Applications 280 8.5 Problems 285 8.6. Simulation Project 287

Chapter 9 Hypothesis Testing 289 9.1. Introduction 290 9.2. General Procedure 290 9.3. Hypothesis Tests of Means 295 9.4. Hypothesis Tests of Variances 302 9.5. Tests of Distributions 308 9.6. Applications 319 9.7. Simulation of Hypothesis Test Assumptions 326 9.8. Problems 328 9.9 Simulation Projects 333

Chapter 10 Analysis of Variance 335 10.1. Introduction 335 10.2. Test of Population Means 336 10.3. Multiple Comparisons in the ANOVA Test 345 10.4. Test of Population Variances 349 10.5. Randomized Block Design 351 10.6 Two-Way Analysis of Variance 357 10.7. Applications 370 10.8. Problems 372 10.9. Simulation Projects 377

Chapter 11 Confidence Intervals and Sample Size Determination 379 11.1. Introduction 379 11.2. General Procedure 380 11.3. Confidence Intervals on Sample Statistics 381 11.4. Sample-Size Determination 384 11.5. Applications 387 11.6. Problems 389 11.7. Simulation Projects 391

Chapter 12 Regression Analysis 393 12.1. Introduction 394 12.2. Correlation Analysis 394 12.3. Introduction to Regression 404 12.4. Principle of Least Squares 409 12.5. Reliability of the Regression Equation 412 12.6. Reliability of Point Estimates of the Regression Coefficients 420 12.7. Confidence Intervals of the Regression Equation 423 12.8. Correlation vs. Regression 428 12.9. Applications of Bivariate Regression Analysis 429 12.10. Simulation and Prediction Models 437 12.11. Problems 439 12.12. Simulation Projects 445

Chapter 13 Multiple and Nonlinear Regression Analysis 447 13.1. Introduction 448 13.2. Correlation Analysis 448 13.3. Multiple Regression Analysis 450 13.4. Polynomial Regression Analysis 459 13.5. Regression Analysis of Power Models 464 13.6. Applications 466 13.7. Simulation in Curvilinear Modeling 478 13.8. Problems 481 13.9. Simulation Projects 484

Chapter 14 Reliability Analysis of Components 485 14.1. Introduction 485 14.2. Time to Failure 486 14.3. Reliability of Components 489 14.4. First-Order Reliability Method 491 14.5. Advanced Second-Moment Method 496 14.6. Simulation Methods 509 14.7. Reliability-Based Design 519 14.8. Application: Structural Reliability of a Pressure Vessel 525 14.9. Problems 530 14.10. Simulation Projects 534

Chapter 15 Reliability and Risk Analysis of Systems 535 15.1. Introduction 535 15.2. Reliability of Systems 537 15.3. Risk Analysis 551 15.4. Risk-Based Decision Analysis 557 15.5. Application: System Reliability of a Post-Tensioned Truss 562 15.6. Problems 565 15.7. Simulation Projects 568

Chapter 16 Bayesian Methods 569 16.1. Introduction 569 16.2. Bayesian Probabilities 570 16.3. Bayesian Estimation of Parameters 575 16.4. Bayesian Statistics 584 16.5. Applications 588 16.6. Problems 591

Appendix A Probability and Statistics Tables 595 A-l. Cumulative Distribution Function of Standard Normal (O(z)) 596 A-2. Critical Values for the Student's t Distribution (tak) 599 A-3. Critical Values for the Chi-Square Distribution (cak = %2

ak) 601 A-4. Critical Values for the F Distribution (faku = /a,v,,v;) 603 A-5. Critical Values for the Pearson Correlation Coefficient for

the Null Hypothesis H0.- p = 0 and Both the One-Tailed Alternative HA: |p| > 0 and the Two-Tailed Alternative HA: p * 0 608

A-6. Uniformly Distributed Random Numbers 609 A-7. Critical Values for the Kolmogorov-Smirnov One-Sample Test 610 A-8. Values of the Gamma Function 611 A-9. Critical Values for the Duncan Multiple Range Test for a 5% Level

of Significance and Selected Degrees of Freedom (dj) and p Groups 612

Appendix B Taylor Series Expansion 613 B-l. Taylor Series 613 B-2. Common Taylor Series 616 B-3. Applications: Taylor Series Expansion of the Square Root 617 B-4. Problems 618

Appendix C Data for Simulation Projects 621 C-l. Stream Erosion Study 621 C-2. Traffic Estimation Study 622 C-3. Water Evaporation Study 623

Index 625