JOHN I. McCOOL, PhD, is Professor of Systems Engineering at Penn State Great Valley School of Graduate Professional Studies. A Fellow of the American Society for Quality, Dr. McCool previously served as principal engineering scientist at SKF Industries Inc., where he conducted corporate as well as federally sponsored research projects with the Wright-Patterson Air Force Base, the Office of Naval Research, the Naval Air Propulsion Center, the Department of Energy, and the Air Force Office of Scientific Research.

• Preface xiii1. Probability 11.1 Sample Spaces and Events 21.2 Mutually Exclusive Events 21.3 Venn Diagrams 31.4 Unions of Events and Joint Probability 41.5 Conditional Probability 61.6 Independence 81.7 Partitions and the Law of Total Probability 91.8 Reliability 121.9 Series Systems 121.10 Parallel Systems 131.11 Complex Systems 151.12 Crosslinked Systems 161.13 Reliability Importance 19References 20Exercises 212. Discrete and Continuous Random Variables 232.1 Probability Distributions 242.2 Functions of a Random Variable 262.3 Jointly Distributed Discrete Random Variables 282.4 Conditional Expectation 322.5 The Binomial Distribution 342.6 The Poisson Distribution 392.7 The Geometric Distribution 412.8 Continuous Random Variables 422.9 Jointly Distributed Continuous Random Variables 512.10 Simulating Samples from Continuous Distributions 522.11 The Normal Distribution 542.12 Distribution of the Sample Mean 602.13 The Lognormal Distribution 662.14 Simple Linear Regression 67References 69Exercises 693. Properties of the Weibull Distribution 733.1 The Weibull Cumulative Distribution Function (CDF) Percentiles Moments and Hazard Function 733.2 The Minima of Weibull Samples 823.3 Transformations 833.4 The Conditional Weibull Distribution 863.5 Quantiles for Order Statistics of a Weibull Sample 893.6 Simulating Weibull Samples 92References 94Exercises 954. Weibull Probability Models 974.1 System Reliability 974.2 Weibull Mixtures 1034.3 P(Y < X) 1054.4 Radial Error 1084.5 Pro Rata Warranty 1104.6 Optimum Age Replacement 1124.7 Renewal Theory 1194.8 Optimum Bidding 1234.9 Optimum Burn-In 1244.10 Spare Parts Provisioning 126References 127Exercises 1285. Estimation in Single Samples 1305.1 Point and Interval Estimation 1305.2 Censoring 1305.3 Estimation Methods 1325.4 Graphical Estimation of Weibull Parameters 1365.5 Maximum Likelihood Estimation 1455.6 ML Estimation for the Weibull Distribution 154References 177Exercises 1796. Sample Size Selection Hypothesis Testing and Goodness of Fit 1806.1 Precision Measure for Maximum Likelihood (ML) Estimates 1806.2 Interval Estimates from Menon’s Method of Estimation 1826.3 Hypothesis Testing—Single Samples 1846.4 Operating Characteristic (OC) Curves for One-Sided Tests of the Weibull Shape Parameter 1886.5 OC Curves for One-Sided Tests on a Weibull Percentile 1916.6 Goodness of Fit 1956.7 Lognormal versus Weibull 204References 210Exercises 2127. The Program Pivotal.exe 2137.1 Relationship among Quantiles 2167.2 Series Systems 2177.3 Confidence Limits on Reliability 2187.4 Using Pivotal.exe for OC Curve Calculations 2217.5 Prediction Intervals 2247.6 Sudden Death Tests 2267.7 Design of Optimal Sudden Death Tests 230References 233Exercises 2348. Inference from Multiple Samples 2358.1 Multiple Weibull Samples 2358.2 Testing the Homogeneity of Shape Parameters 2368.3 Estimating the Common Shape Parameter 2388.4 Interval Estimation of a Percentile 2448.5 Testing Whether the Scale Parameters Are Equal 2498.6 Multiple Comparison Tests for Differences in Scale Parameters 2578.7 An Alternative Multiple Comparison Test for Percentiles 2598.8 The Program Multi-Weibull.exe 2618.9 Inference on P (Y < X) 266References 274Exercises 2749. Weibull Regression 2769.1 The Power Law Model 2769.2 ML Estimation 2789.3 Example 2799.4 Pivotal Functions 2809.5 Confidence Intervals 2819.6 Testing the Power Law Model 2819.7 Monte Carlo Results 2829.8 Example Concluded 2859.9 Approximating u* at Other Stress Levels 2879.10 Precision 2899.11 Stress Levels in Different Proportions Than Tabulated 2899.12 Discussion 2919.13 The Disk Operating System (DOS) Program REGEST 291References 296Exercises 29610. The Three-Parameter Weibull Distribution 29810.1 The Model 29810.2 Estimation and Inference for the Weibull Location Parameter 30010.3 Testing the Two- versus Three-Parameter Weibull Distribution 30110.4 Power of the Test 30210.5 Interval Estimation 30210.6 Input and Output Screens of LOCEST.exe 30710.7 The Program LocationPivotal.exe 30910.8 Simulated Example 311References 311Exercises 31211 Factorial Experiments with Weibull Response 31311.1 Introduction 31311.2 The Multiplicative Model 31411.3 Data 31711.4 Estimation 31711.5 Test for the Appropriate Model 31911.6 Monte Carlo Results 32011.7 The DOS Program TWOWAY 32011.8 Illustration of the Influence of Factor Effects on the Shape Parameter Estimates 32011.9 Numerical Examples 327References 331Exercises 332Index 333

JOHN I. McCOOL, PhD, is Professor of Systems Engineering at Penn State Great Valley School of Graduate Professional Studies. A Fellow of the American Society for Quality, Dr. McCool previously served as principal engineering scientist at SKF Industries Inc., where he conducted corporate as well as federally sponsored research projects with the Wright-Patterson Air Force Base, the Office of Naval Research, the Naval Air Propulsion Center, the Department of Energy, and the Air Force Office of Scientific Research.

• Preface xiii1. Probability 11.1 Sample Spaces and Events 21.2 Mutually Exclusive Events 21.3 Venn Diagrams 31.4 Unions of Events and Joint Probability 41.5 Conditional Probability 61.6 Independence 81.7 Partitions and the Law of Total Probability 91.8 Reliability 121.9 Series Systems 121.10 Parallel Systems 131.11 Complex Systems 151.12 Crosslinked Systems 161.13 Reliability Importance 19References 20Exercises 212. Discrete and Continuous Random Variables 232.1 Probability Distributions 242.2 Functions of a Random Variable 262.3 Jointly Distributed Discrete Random Variables 282.4 Conditional Expectation 322.5 The Binomial Distribution 342.6 The Poisson Distribution 392.7 The Geometric Distribution 412.8 Continuous Random Variables 422.9 Jointly Distributed Continuous Random Variables 512.10 Simulating Samples from Continuous Distributions 522.11 The Normal Distribution 542.12 Distribution of the Sample Mean 602.13 The Lognormal Distribution 662.14 Simple Linear Regression 67References 69Exercises 693. Properties of the Weibull Distribution 733.1 The Weibull Cumulative Distribution Function (CDF) Percentiles Moments and Hazard Function 733.2 The Minima of Weibull Samples 823.3 Transformations 833.4 The Conditional Weibull Distribution 863.5 Quantiles for Order Statistics of a Weibull Sample 893.6 Simulating Weibull Samples 92References 94Exercises 954. Weibull Probability Models 974.1 System Reliability 974.2 Weibull Mixtures 1034.3 P(Y < X) 1054.4 Radial Error 1084.5 Pro Rata Warranty 1104.6 Optimum Age Replacement 1124.7 Renewal Theory 1194.8 Optimum Bidding 1234.9 Optimum Burn-In 1244.10 Spare Parts Provisioning 126References 127Exercises 1285. Estimation in Single Samples 1305.1 Point and Interval Estimation 1305.2 Censoring 1305.3 Estimation Methods 1325.4 Graphical Estimation of Weibull Parameters 1365.5 Maximum Likelihood Estimation 1455.6 ML Estimation for the Weibull Distribution 154References 177Exercises 1796. Sample Size Selection Hypothesis Testing and Goodness of Fit 1806.1 Precision Measure for Maximum Likelihood (ML) Estimates 1806.2 Interval Estimates from Menon’s Method of Estimation 1826.3 Hypothesis Testing—Single Samples 1846.4 Operating Characteristic (OC) Curves for One-Sided Tests of the Weibull Shape Parameter 1886.5 OC Curves for One-Sided Tests on a Weibull Percentile 1916.6 Goodness of Fit 1956.7 Lognormal versus Weibull 204References 210Exercises 2127. The Program Pivotal.exe 2137.1 Relationship among Quantiles 2167.2 Series Systems 2177.3 Confidence Limits on Reliability 2187.4 Using Pivotal.exe for OC Curve Calculations 2217.5 Prediction Intervals 2247.6 Sudden Death Tests 2267.7 Design of Optimal Sudden Death Tests 230References 233Exercises 2348. Inference from Multiple Samples 2358.1 Multiple Weibull Samples 2358.2 Testing the Homogeneity of Shape Parameters 2368.3 Estimating the Common Shape Parameter 2388.4 Interval Estimation of a Percentile 2448.5 Testing Whether the Scale Parameters Are Equal 2498.6 Multiple Comparison Tests for Differences in Scale Parameters 2578.7 An Alternative Multiple Comparison Test for Percentiles 2598.8 The Program Multi-Weibull.exe 2618.9 Inference on P (Y < X) 266References 274Exercises 2749. Weibull Regression 2769.1 The Power Law Model 2769.2 ML Estimation 2789.3 Example 2799.4 Pivotal Functions 2809.5 Confidence Intervals 2819.6 Testing the Power Law Model 2819.7 Monte Carlo Results 2829.8 Example Concluded 2859.9 Approximating u* at Other Stress Levels 2879.10 Precision 2899.11 Stress Levels in Different Proportions Than Tabulated 2899.12 Discussion 2919.13 The Disk Operating System (DOS) Program REGEST 291References 296Exercises 29610. The Three-Parameter Weibull Distribution 29810.1 The Model 29810.2 Estimation and Inference for the Weibull Location Parameter 30010.3 Testing the Two- versus Three-Parameter Weibull Distribution 30110.4 Power of the Test 30210.5 Interval Estimation 30210.6 Input and Output Screens of LOCEST.exe 30710.7 The Program LocationPivotal.exe 30910.8 Simulated Example 311References 311Exercises 31211 Factorial Experiments with Weibull Response 31311.1 Introduction 31311.2 The Multiplicative Model 31411.3 Data 31711.4 Estimation 31711.5 Test for the Appropriate Model 31911.6 Monte Carlo Results 32011.7 The DOS Program TWOWAY 32011.8 Illustration of the Influence of Factor Effects on the Shape Parameter Estimates 32011.9 Numerical Examples 327References 331Exercises 332Index 333

JOHN I. McCOOL, PhD, is Professor of Systems Engineering at Penn State Great Valley School of Graduate Professional Studies. A Fellow of the American Society for Quality, Dr. McCool previously served as principal engineering scientist at SKF Industries Inc., where he conducted corporate as well as federally sponsored research projects with the Wright-Patterson Air Force Base, the Office of Naval Research, the Naval Air Propulsion Center, the Department of Energy, and the Air Force Office of Scientific Research.

• Preface xiii1. Probability 11.1 Sample Spaces and Events 21.2 Mutually Exclusive Events 21.3 Venn Diagrams 31.4 Unions of Events and Joint Probability 41.5 Conditional Probability 61.6 Independence 81.7 Partitions and the Law of Total Probability 91.8 Reliability 121.9 Series Systems 121.10 Parallel Systems 131.11 Complex Systems 151.12 Crosslinked Systems 161.13 Reliability Importance 19References 20Exercises 212. Discrete and Continuous Random Variables 232.1 Probability Distributions 242.2 Functions of a Random Variable 262.3 Jointly Distributed Discrete Random Variables 282.4 Conditional Expectation 322.5 The Binomial Distribution 342.6 The Poisson Distribution 392.7 The Geometric Distribution 412.8 Continuous Random Variables 422.9 Jointly Distributed Continuous Random Variables 512.10 Simulating Samples from Continuous Distributions 522.11 The Normal Distribution 542.12 Distribution of the Sample Mean 602.13 The Lognormal Distribution 662.14 Simple Linear Regression 67References 69Exercises 693. Properties of the Weibull Distribution 733.1 The Weibull Cumulative Distribution Function (CDF) Percentiles Moments and Hazard Function 733.2 The Minima of Weibull Samples 823.3 Transformations 833.4 The Conditional Weibull Distribution 863.5 Quantiles for Order Statistics of a Weibull Sample 893.6 Simulating Weibull Samples 92References 94Exercises 954. Weibull Probability Models 974.1 System Reliability 974.2 Weibull Mixtures 1034.3 P(Y < X) 1054.4 Radial Error 1084.5 Pro Rata Warranty 1104.6 Optimum Age Replacement 1124.7 Renewal Theory 1194.8 Optimum Bidding 1234.9 Optimum Burn-In 1244.10 Spare Parts Provisioning 126References 127Exercises 1285. Estimation in Single Samples 1305.1 Point and Interval Estimation 1305.2 Censoring 1305.3 Estimation Methods 1325.4 Graphical Estimation of Weibull Parameters 1365.5 Maximum Likelihood Estimation 1455.6 ML Estimation for the Weibull Distribution 154References 177Exercises 1796. Sample Size Selection Hypothesis Testing and Goodness of Fit 1806.1 Precision Measure for Maximum Likelihood (ML) Estimates 1806.2 Interval Estimates from Menon’s Method of Estimation 1826.3 Hypothesis Testing—Single Samples 1846.4 Operating Characteristic (OC) Curves for One-Sided Tests of the Weibull Shape Parameter 1886.5 OC Curves for One-Sided Tests on a Weibull Percentile 1916.6 Goodness of Fit 1956.7 Lognormal versus Weibull 204References 210Exercises 2127. The Program Pivotal.exe 2137.1 Relationship among Quantiles 2167.2 Series Systems 2177.3 Confidence Limits on Reliability 2187.4 Using Pivotal.exe for OC Curve Calculations 2217.5 Prediction Intervals 2247.6 Sudden Death Tests 2267.7 Design of Optimal Sudden Death Tests 230References 233Exercises 2348. Inference from Multiple Samples 2358.1 Multiple Weibull Samples 2358.2 Testing the Homogeneity of Shape Parameters 2368.3 Estimating the Common Shape Parameter 2388.4 Interval Estimation of a Percentile 2448.5 Testing Whether the Scale Parameters Are Equal 2498.6 Multiple Comparison Tests for Differences in Scale Parameters 2578.7 An Alternative Multiple Comparison Test for Percentiles 2598.8 The Program Multi-Weibull.exe 2618.9 Inference on P (Y < X) 266References 274Exercises 2749. Weibull Regression 2769.1 The Power Law Model 2769.2 ML Estimation 2789.3 Example 2799.4 Pivotal Functions 2809.5 Confidence Intervals 2819.6 Testing the Power Law Model 2819.7 Monte Carlo Results 2829.8 Example Concluded 2859.9 Approximating u* at Other Stress Levels 2879.10 Precision 2899.11 Stress Levels in Different Proportions Than Tabulated 2899.12 Discussion 2919.13 The Disk Operating System (DOS) Program REGEST 291References 296Exercises 29610. The Three-Parameter Weibull Distribution 29810.1 The Model 29810.2 Estimation and Inference for the Weibull Location Parameter 30010.3 Testing the Two- versus Three-Parameter Weibull Distribution 30110.4 Power of the Test 30210.5 Interval Estimation 30210.6 Input and Output Screens of LOCEST.exe 30710.7 The Program LocationPivotal.exe 30910.8 Simulated Example 311References 311Exercises 31211 Factorial Experiments with Weibull Response 31311.1 Introduction 31311.2 The Multiplicative Model 31411.3 Data 31711.4 Estimation 31711.5 Test for the Appropriate Model 31911.6 Monte Carlo Results 32011.7 The DOS Program TWOWAY 32011.8 Illustration of the Influence of Factor Effects on the Shape Parameter Estimates 32011.9 Numerical Examples 327References 331Exercises 332Index 333