Instructor Solution Manueal for PROBABILITY & STATISTICS FOR ENGINEERS & SCIENTISTS -- 8th ed. (KEYING YE, SHARON MYERS)

Contents
1 Introduction to Statistics and Data Analysis 1
2 Probability 11
3 Random Variables and Probability Distributions 29
4 Mathematical Expectation 45
5 Some Discrete Probability Distributions 59
6 Some Continuous Probability Distributions 71
7 Functions of Random Variables 85
8 Fundamental Sampling Distributions and Data Descriptions 91
9 One- and Two-Sample Estimation Problems 103
10 One- and Two-Sample Tests of Hypotheses 121
11 Simple Linear Regression and Correlation 149
12 Multiple Linear Regression and Certain Nonlinear Regression Models 171
13 One-Factor Experiments: General 185
14 Factorial Experiments (Two or More Factors) 213
15 2 k Factorial Experiments and Fractions 237
16 Nonparametric Statistics 257
17 Statistical Quality Control 273
18 Bayesian Statistics 277

Introduction to Probability and Statistics for Engineers and Scientists
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Applied Statistics Using SPSS, STATISTICA, MATLAB and R

Contents
Preface to the Second Edition xv
Preface to the First Edition xvii
Symbols and Abbreviations xix
1 Introduction 1
1.1 Deterministic Data and Random Data.........................................................1
1.2 Population, Sample and Statistics ...............................................................5
1.3 Random Variables.......................................................................................8
1.4 Probabilities and Distributions..................................................................10
1.4.1 Discrete Variables .......................................................................10
1.4.2 Continuous Variables ..................................................................12
1.5 Beyond a Reasonable Doubt... ..................................................................13
1.6 Statistical Significance and Other Significances.......................................17
1.7 Datasets.....................................................................................................19
1.8 Software Tools ..........................................................................................19
1.8.1 SPSS and STATISTICA..............................................................20
1.8.2 MATLAB and R..........................................................................22
2 Presenting and Summarising the Data 29
2.1 Preliminaries .............................................................................................29
2.1.1 Reading in the Data .....................................................................29
2.1.2 Operating with the Data...............................................................34
2.2 Presenting the Data ...................................................................................39
2.2.1 Counts and Bar Graphs................................................................40
2.2.2 Frequencies and Histograms........................................................47
2.2.3 Multivariate Tables, Scatter Plots and 3D Plots ..........................52
2.2.4 Categorised Plots .........................................................................56
2.3 Summarising the Data...............................................................................58
2.3.1 Measures of Location ..................................................................58
2.3.2 Measures of Spread .....................................................................62
2.3.3 Measures of Shape.......................................................................64
2.3.4 Measures of Association for Continuous Variables.....................66
2.3.5 Measures of Association for Ordinal Variables...........................69
2.3.6 Measures of Association for Nominal Variables.........................73
Exercises.................................................................................................................77
3 Estimating Data Parameters 81
3.1 Point Estimation and Interval Estimation..................................................81
3.2 Estimating a Mean ....................................................................................85
3.3 Estimating a Proportion ............................................................................92
3.4 Estimating a Variance ...............................................................................95
3.5 Estimating a Variance Ratio......................................................................97
3.6 Bootstrap Estimation.................................................................................99
Exercises...............................................................................................................107
4 Parametric Tests of Hypotheses 111
4.1 Hypothesis Test Procedure......................................................................111
4.2 Test Errors and Test Power.....................................................................115
4.3 Inference on One Population...................................................................121
4.3.1 Testing a Mean ..........................................................................121
4.3.2 Testing a Variance.....................................................................125
4.4 Inference on Two Populations ................................................................126
4.4.1 Testing a Correlation .................................................................126
4.4.2 Comparing Two Variances........................................................129
4.4.3 Comparing Two Means .............................................................132
4.5 Inference on More than Two Populations..............................................141
4.5.1 Introduction to the Analysis of Variance...................................141
4.5.2 One-Way ANOVA ....................................................................143
4.5.3 Two-Way ANOVA ...................................................................156
Exercises...............................................................................................................166
5 Non-Parametric Tests of Hypotheses 171
5.1 Inference on One Population...................................................................172
5.1.1 The Runs Test............................................................................172
5.1.2 The Binomial Test .....................................................................174
5.1.3 The Chi-Square Goodness of Fit Test .......................................179
5.1.4 The Kolmogorov-Smirnov Goodness of Fit Test ......................183
5.1.5 The Lilliefors Test for Normality ..............................................187
5.1.6 The Shapiro-Wilk Test for Normality .......................................187
5.2 Contingency Tables.................................................................................189
5.2.1 The 2×2 Contingency Table ......................................................189
5.2.2 The rxc Contingency Table .......................................................193
5.2.3 The Chi-Square Test of Independence ......................................195
5.2.4 Measures of Association Revisited............................................197
5.3 Inference on Two Populations ................................................................200
5.3.1 Tests for Two Independent Samples..........................................201
5.3.2 Tests for Two Paired Samples ...................................................205
5.4 Inference on More Than Two Populations..............................................212
5.4.1 The Kruskal-Wallis Test for Independent Samples...................212
5.4.2 The Friedmann Test for Paired Samples ...................................215
5.4.3 The Cochran Q test....................................................................217
Exercises...............................................................................................................218
6 Statistical Classification 223
6.1 Decision Regions and Functions.............................................................223
6.2 Linear Discriminants...............................................................................225
6.2.1 Minimum Euclidian Distance Discriminant ..............................225
6.2.2 Minimum Mahalanobis Distance Discriminant.........................228
6.3 Bayesian Classification ...........................................................................234
6.3.1 Bayes Rule for Minimum Risk..................................................234
6.3.2 Normal Bayesian Classification ................................................240
6.3.3 Dimensionality Ratio and Error Estimation...............................243
6.4 The ROC Curve ......................................................................................246
6.5 Feature Selection.....................................................................................253
6.6 Classifier Evaluation...............................................................................256
6.7 Tree Classifiers .......................................................................................259
Exercises...............................................................................................................268
7 Data Regression 271
7.1 Simple Linear Regression .......................................................................272
7.1.1 Simple Linear Regression Model ..............................................272
7.1.2 Estimating the Regression Function ..........................................273
7.1.3 Inferences in Regression Analysis.............................................279
7.1.4 ANOVA Tests ...........................................................................285
7.2 Multiple Regression ................................................................................289
7.2.1 General Linear Regression Model.............................................289
7.2.2 General Linear Regression in Matrix Terms .............................289
7.2.3 Multiple Correlation ..................................................................292
7.2.4 Inferences on Regression Parameters ........................................294
7.2.5 ANOVA and Extra Sums of Squares.........................................296
7.2.6 Polynomial Regression and Other Models ................................300
7.3 Building and Evaluating the Regression Model......................................303
7.3.1 Building the Model....................................................................303
7.3.2 Evaluating the Model ................................................................306
7.3.3 Case Study.................................................................................308
7.4 Regression Through the Origin...............................................................314
7.5 Ridge Regression ....................................................................................316
7.6 Logit and Probit Models .........................................................................322
Exercises...............................................................................................................327
8 Data Structure Analysis 329
8.1 Principal Components .............................................................................329
8.2 Dimensional Reduction...........................................................................337
8.3 Principal Components of Correlation Matrices.......................................339
8.4 Factor Analysis .......................................................................................347
Exercises...............................................................................................................350
9 Survival Analysis 353
9.1 Survivor Function and Hazard Function .................................................353
9.2 Non-Parametric Analysis of Survival Data.............................................354
9.2.1 The Life Table Analysis ............................................................354
9.2.2 The Kaplan-Meier Analysis.......................................................359
9.2.3 Statistics for Non-Parametric Analysis......................................362
9.3 Comparing Two Groups of Survival Data ..............................................364
9.4 Models for Survival Data........................................................................367
9.4.1 The Exponential Model .............................................................367
9.4.2 The Weibull Model....................................................................369
9.4.3 The Cox Regression Model .......................................................371
Exercises...............................................................................................................373
10 Directional Data 375
10.1 Representing Directional Data ................................................................375
10.2 Descriptive Statistics...............................................................................380
10.3 The von Mises Distributions ...................................................................383
10.4 Assessing the Distribution of Directional Data.......................................387
10.4.1 Graphical Assessment of Uniformity ........................................387
10.4.2 The Rayleigh Test of Uniformity ..............................................389
10.4.3 The Watson Goodness of Fit Test .............................................392
10.4.4 Assessing the von Misesness of Spherical Distributions...........393
10.5 Tests on von Mises Distributions............................................................395
10.5.1 One-Sample Mean Test .............................................................395
10.5.2 Mean Test for Two Independent Samples .................................396
10.6 Non-Parametric Tests..............................................................................397
10.6.1 The Uniform Scores Test for Circular Data...............................397
10.6.2 The Watson Test for Spherical Data..........................................398
10.6.3 Testing Two Paired Samples .....................................................399
Exercises...............................................................................................................400
Appendix A - Short Survey on Probability Theory 403
A.1 Basic Notions ..........................................................................................403
A.1.1 Events and Frequencies .............................................................403
A.1.2 Probability Axioms....................................................................404
A.2 Conditional Probability and Independence .............................................406
A.2.1 Conditional Probability and Intersection Rule...........................406
A.2.2 Independent Events ...................................................................406
A.3 Compound Experiments..........................................................................408
A.4 Bayes’ Theorem ......................................................................................409
A.5 Random Variables and Distributions ......................................................410
A.5.1 Definition of Random Variable .................................................410
A.5.2 Distribution and Density Functions ...........................................411
A.5.3 Transformation of a Random Variable ......................................413
A.6 Expectation, Variance and Moments ......................................................414
A.6.1 Definitions and Properties .........................................................414
A.6.2 Moment-Generating Function ...................................................417
A.6.3 Chebyshev Theorem..................................................................418
A.7 The Binomial and Normal Distributions.................................................418
A.7.1 The Binomial Distribution.........................................................418
A.7.2 The Laws of Large Numbers .....................................................419
A.7.3 The Normal Distribution ...........................................................420
A.8 Multivariate Distributions .......................................................................422
A.8.1 Definitions .................................................................................422
A.8.2 Moments....................................................................................425
A.8.3 Conditional Densities and Independence...................................425
A.8.4 Sums of Random Variables .......................................................427
A.8.5 Central Limit Theorem ..............................................................428
Appendix B - Distributions 431
B.1 Discrete Distributions .............................................................................431
B.1.1 Bernoulli Distribution................................................................431
B.1.2 Uniform Distribution .................................................................432
B.1.3 Geometric Distribution..............................................................433
B.1.4 Hypergeometric Distribution.....................................................434
B.1.5 Binomial Distribution................................................................435
B.1.6 Multinomial Distribution...........................................................436
B.1.7 Poisson Distribution ..................................................................438
B.2 Continuous Distributions ........................................................................439
B.2.1 Uniform Distribution .................................................................439
B.2.2 Normal Distribution...................................................................441
B.2.3 Exponential Distribution............................................................442
B.2.4 Weibull Distribution..................................................................444
B.2.5 Gamma Distribution ..................................................................445
B.2.6 Beta Distribution .......................................................................446
B.2.7 Chi-Square Distribution.............................................................448
B.2.8 Student’s t Distribution..............................................................449
B.2.9 F Distribution ...........................................................................451
B.2.10 Von Mises Distributions............................................................452
Appendix C - Point Estimation 455
C.1 Definitions...............................................................................................455
C.2 Estimation of Mean and Variance...........................................................457
Appendix D - Tables 459
D.1 Binomial Distribution .............................................................................459
D.2 Normal Distribution ................................................................................465
D.3 Student´s t Distribution ...........................................................................466
D.4 Chi-Square Distribution ..........................................................................467
D.5 Critical Values for the F Distribution .....................................................468
Appendix E - Datasets 469
E.1 Breast Tissue...........................................................................................469
E.2 Car Sale...................................................................................................469
E.3 Cells ........................................................................................................470
E.4 Clays .......................................................................................................470
E.5 Cork Stoppers..........................................................................................471
E.6 CTG ........................................................................................................472
E.7 Culture ....................................................................................................473
E.8 Fatigue ....................................................................................................473
E.9 FHR.........................................................................................................474
E.10 FHR-Apgar .............................................................................................474
E.11 Firms .......................................................................................................475
E.12 Flow Rate................................................................................................475
E.13 Foetal Weight..........................................................................................475
E.14 Forest Fires..............................................................................................476
E.15 Freshmen.................................................................................................476
E.16 Heart Valve .............................................................................................477
E.17 Infarct......................................................................................................478
E.18 Joints .......................................................................................................478
E.19 Metal Firms.............................................................................................479
E.20 Meteo ......................................................................................................479
E.21 Moulds ....................................................................................................479
E.22 Neonatal ..................................................................................................480
E.23 Programming...........................................................................................480
E.24 Rocks ......................................................................................................481
E.25 Signal & Noise........................................................................................481
E.26 Soil Pollution ..........................................................................................482
E.27 Stars ........................................................................................................482
E.28 Stock Exchange.......................................................................................483
E.29 VCG ........................................................................................................484
E.30 Wave .......................................................................................................484
E.31 Weather...................................................................................................484
E.32 Wines ......................................................................................................485
Appendix F - Tools 487
F.1 MATLAB Functions...............................................................................487
F.2 R Functions .............................................................................................488
F.3 Tools EXCEL File ..................................................................................489
F.4 SCSize Program ......................................................................................489
References 491
Index 499

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Applied Statistics and Probability for Engineers [Solution Manual]

The purpose of this Student Solutions Manual is to provide you with additional help in understanding the problem-solving processes presented in Applied Statistics and Probability for Engineers. The Applied Statistics text includes a section entitled “Answers to Selected Exercises,” which contains the final answers to most odd-numbered exercises in the book. Within the text, problems with an answer available are indicated by the exercise number enclosed in a box.

This Student Solutions Manual provides complete worked-out solutions to a subset of the problems included in the “Answers to Selected Exercises.” If you are having difficulty reaching the final answer provided in the text, the complete solution will help you determine the correct way to solve the problem.

Those problems with a complete solution available are indicated in the “Answers to
Selected Exercises,” again by a box around the exercise number. The complete solutions to this subset of problems may also be accessed by going directly to this Student Solutions Manual.


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Solutions Manual A First Course in Probability

Sheldon M. Ross, "A First Course in Probability, Fifth Edition"
Prentice Hall College | 1997 | ISBN: 0137463146 |
This market-leading introduction to probability features exceptionally clear explanations of the mathematics of probability theory and explores its many diverse applications through numerous interesting and motivational examples. The outstanding problem sets are a hallmark feature of this book. Provides clear, complete explanations to fully explain mathematical concepts. Features subsections on the probabilistic method and the maximum-minimums identity. Includes many new examples relating to DNA matching, utility, finance, and applications of the probabilistic method. Features an intuitive treatment of probability—intuitive explanations follow many examples. The Probability Models Disk included with each copy of the book, contains six probability models that are referenced in the book and allow readers to quickly and easily perform calculations and simulations.


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Random Networks for Communication

Contents
page ix
Preface
List of notation xi
1 Introduction 1
1.1 Discrete network models 3
1.1.1 The random tree 3
1.1.2 The random grid 5
1.2 Continuum network models 6
1.2.1 Poisson processes 6
1.2.2 Nearest neighbour networks 10
1.2.3 Poisson random connection networks 11
1.2.4 Boolean model networks 12
1.2.5 Interference limited networks 13
1.3 Information-theoretic networks 14
1.4 Historical notes and further reading 15
2 Phase transitions in infinite networks 17
2.1 The random tree; infinite growth 17
2.2 The random grid; discrete percolation 21
2.3 Dependencies 29
2.4 Nearest neighbours; continuum percolation 31
2.5 Random connection model 37
2.6 Boolean model 48
2.7 Interference limited networks 51
2.7.1 Mapping on a square lattice 56
2.7.2 Percolation on the square lattice 58
2.7.3 Percolation of the interference model 62
2.7.4 Bound on the percolation region 63
2.8 Historical notes and further reading 66
3 Connectivity of finite networks 69
3.1 Preliminaries: modes of convergence and Poisson approximation 69
3.2 The random grid 71
3.2.1 Almost connectivity 71
3.2.2 Full connectivity 72
3.3 Boolean model 77
3.3.1 Almost connectivity 78
3.3.2 Full connectivity 81
3.4 Nearest neighbours; full connectivity 88
3.5 Critical node lifetimes 92
3.6 A central limit theorem 98
3.7 Historical notes and further reading 98
4 More on phase transitions 100
4.1 Preliminaries: Harris–FKG Inequality 100
4.2 Uniqueness of the infinite cluster 101
4.3 Cluster size distribution and crossing paths 107
4.4 Threshold behaviour of fixed size networks 114
4.5 Historical notes and further reading 119
5 Information flow in random networks 121
5.1 Information-theoretic preliminaries 121
5.1.1 Channel capacity 122
5.1.2 Additive Gaussian channel 124
5.1.3 Communication with continuous time signals 127
5.1.4 Information-theoretic random networks 129
5.2 Scaling limits; single source–destination pair 131
5.3 Multiple source–destination pairs; lower bound 136
5.3.1 The highway 138
5.3.2 Capacity of the highway 139
5.3.3 Routing protocol 142
5.4 Multiple source–destination pairs; information-theoretic upper
bounds 146
5.4.1 Exponential attenuation case 148
5.4.2 Power law attenuation case 151
5.5 Historical notes and further reading 155
6 Navigation in random networks 157
6.1 Highway discovery 157
6.2 Discrete short-range percolation (large worlds) 159
6.3 Discrete long-range percolation (small worlds) 161
6.3.1 Chemical distance, diameter, and navigation length 162
6.3.2 More on navigation length 167
6.4 Continuum long-range percolation (small worlds) 171
6.5 The role of scale invariance in networks 181
6.6 Historical notes and further reading 182
Appendix 185
A.1 Landau’s order notation 185
A.2 Stirling’s formula 185
A.3 Ergodicity and the ergodic theorem 185
A.4 Deviations from the mean 187
A.5 The Cauchy–Schwartz inequality 188
A.6 The singular value decomposition 189
References 190
Index 194

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Regression Linear Models in Statistics

Contents
1. Linear Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Correlation version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Large-sample limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The origins of regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Applications of regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Maximum Likelihood and Least Squares . . . . . . . . . . . . . . . . . . . . . 21
1.7 Sums of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.8 Two regressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2. The Analysis of Variance (ANOVA) . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 The Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Change of variable formula and Jacobians . . . . . . . . . . . . . . . . . . . 36
2.3 The Fisher F-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Normal sample mean and sample variance . . . . . . . . . . . . . . . . . . . 39
2.6 One-Way Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7 Two-Way ANOVA; No Replications . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.8 Two-Way ANOVA: Replications and Interaction . . . . . . . . . . . . . . 52
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3. Multiple Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 The Normal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Solution of the Normal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 Properties of Least-Squares Estimators . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Sum-of-Squares Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.1 Coefficient of determination. . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Chi-Square Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.1 Idempotence, Trace and Rank . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.2 Quadratic forms in normal variates . . . . . . . . . . . . . . . . . . . 82
3.5.3 Sums of Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.6 Orthogonal Projections and Pythagoras’s Theorem . . . . . . . . . . . 85
3.7 Worked examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4. Further Multilinear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1 Polynomial Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1.1 The Principle of Parsimony . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.1.2 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.3 Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 105
4.4 The Multinormal Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4.1 Estimation for the multivariate normal . . . . . . . . . . . . . . . . 113
4.5 Conditioning and Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.6 Mean-square prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.7 Generalised least squares and weighted regression . . . . . . . . . . . . . 123
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5. Adding additional covariates and the Analysis
of Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.1 Introducing further explanatory variables . . . . . . . . . . . . . . . . . . . . 129
5.1.1 Orthogonal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2 ANCOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2.1 Nested Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6. Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.1 Minimisation Under Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 Sum-of-Squares Decomposition and F-Test . . . . . . . . . . . . . . . . . . . 152
6.3 Applications: Sequential Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.3.1 Forward selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.3.2 Backward selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3.3 Stepwise regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7. Model Checking and Transformation of Data . . . . . . . . . . . . . . . 163
7.1 Deviations from Standard Assumptions . . . . . . . . . . . . . . . . . . . . . 163
7.2 Transformation of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.3 Variance-Stabilising Transformations . . . . . . . . . . . . . . . . . . . . . . . . 171
7.4 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8. Generalised Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.2 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.2.1 Statistical testing and model comparisons . . . . . . . . . . . . . 185
8.2.2 Analysis of residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.2.3 Athletics times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.3 Binary models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.4 Count data, contingency tables and log-linear models . . . . . . . . . 193
8.5 Over-dispersion and the Negative Binomial Distribution . . . . . . . 197
8.5.1 Practical applications: Analysis of over-dispersed models
in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
9. Other topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.1 Mixed models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.1.1 Mixed models and Generalised Least Squares . . . . . . . . . . 206
9.2 Non-parametric regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
9.2.1 Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
9.3 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
9.3.1 Optimality criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
9.3.2 Incomplete designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
9.4 Time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
9.4.1 Cointegration and spurious regression . . . . . . . . . . . . . . . . . 220
9.5 Survival analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9.5.1 Proportional hazards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
9.6 p >> n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Dramatis Personae: Who did what when . . . . . . . . . . . . . . . . . . . . . . . 269
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

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Time Series Analysis and Its Applications: With R Examples

Contents
1 Characteristics of Time Series 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Nature of Time Series Data . . . . . . . . . . . . . . . . . 4
1.3 Time Series Statistical Models . . . . . . . . . . . . . . . . . . . 11
1.4 Measures of Dependence: Autocorrelation
and Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Stationary Time Series . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Estimation of Correlation . . . . . . . . . . . . . . . . . . . . . 29
1.7 Vector-Valued andMultidimensional Series . . . . . . . . . . . 34
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Time Series Regression and Exploratory Data Analysis 48
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2 Classical Regression in the Time Series Context . . . . . . . . . 49
2.3 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . 57
2.4 Smoothing in the Time Series Context . . . . . . . . . . . . . . 71
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3 ARIMA Models 84
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2 Autoregressive Moving Average Models . . . . . . . . . . . . . 85
3.3 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4 Autocorrelation and Partial Autocorrelation Functions . . . . . 103
3.5 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.6 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.7 Integrated Models for Nonstationary Data . . . . . . . . . . . . 140
3.8 Building ARIMA Models . . . . . . . . . . . . . . . . . . . . . 143
3.9 Multiplicative Seasonal ARIMA Models . . . . . . . . . . . . . 154
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4 Spectral Analysis and Filtering 174
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.2 Cyclical Behavior and Periodicity . . . . . . . . . . . . . . . . . 176
4.3 The Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . 181
4.4 Periodogramand Discrete Fourier Transform . . . . . . . . . . 187
4.5 Nonparametric Spectral Estimation . . . . . . . . . . . . . . . . 197
4.6 Multiple Series and Cross-Spectra . . . . . . . . . . . . . . . . . 215
4.7 Linear Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.8 Parametric Spectral Estimation . . . . . . . . . . . . . . . . . . 228
4.9 Dynamic Fourier Analysis andWavelets . . . . . . . . . . . . . 232
4.10 Lagged Regression Models . . . . . . . . . . . . . . . . . . . . 245
4.11 Signal Extraction and Optimum Filtering . . . . . . . . . . . . 251
4.12 Spectral Analysis ofMultidimensional Series . . . . . . . . . . . 256
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
5 Additional Time Domain Topics 271
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
5.2 LongMemory ARMA and Fractional Differencing . . . . . . . 271
5.3 GARCHModels . . . . . . . . . . . . . . . . . . . . . . . . . . 280
5.4 ThresholdModels . . . . . . . . . . . . . . . . . . . . . . . . . . 289
5.5 Regression with Autocorrelated Errors . . . . . . . . . . . . . . 293
5.6 Lagged Regression: Transfer Function Modeling . . . . . . . . . 295
5.7 Multivariate ARMAXModels . . . . . . . . . . . . . . . . . . . 302
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
6 State-Space Models 324
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
6.2 Filtering, Smoothing, and Forecasting . . . . . . . . . . . . . . 330
6.3 MaximumLikelihood Estimation . . . . . . . . . . . . . . . . . 339
6.4 Missing Data Modifications . . . . . . . . . . . . . . . . . . . . 348
6.5 StructuralModels: Signal Extraction and Forecasting . . . . . 352
6.6 ARMAX Models in State-Space Form . . . . . . . . . . . . . . 355
6.7 Bootstrapping State-Space Models . . . . . . . . . . . . . . . . 357
6.8 Dynamic LinearModels with Switching . . . . . . . . . . . . . 362
6.9 Nonlinear and Non-normal State-Space
Models UsingMonte CarloMethods . . . . . . . . . . . . . . . 376
6.10 Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . 388
6.11 State-Space and ARMAX Models for
Longitudinal Data Analysis . . . . . . . . . . . . . . . . . . . . 394
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
7 Statistical Methods in the Frequency Domain 412
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
7.2 SpectralMatrices and Likelihood Functions . . . . . . . . . . . 416
7.3 Regression for Jointly Stationary Series . . . . . . . . . . . . . 417
7.4 Regression with Deterministic Inputs . . . . . . . . . . . . . . . 426
7.5 Random Coefficient Regression . . . . . . . . . . . . . . . . . . 434
7.6 Analysis of Designed Experiments . . . . . . . . . . . . . . . . 438
7.7 Discrimination and Cluster Analysis . . . . . . . . . . . . . . . 449
7.8 Principal Components and Factor Analysis . . . . . . . . . . . 464
7.9 The Spectral Envelope . . . . . . . . . . . . . . . . . . . . . . . 479
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
Appendix A: Large Sample Theory 501
A.1 ConvergenceModes . . . . . . . . . . . . . . . . . . . . . . . . 501
A.2 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . . 509
A.3 TheMean and Autocorrelation Functions . . . . . . . . . . . . 513
Appendix B: Time Domain Theory 522
B.1 Hilbert Spaces and the Projection Theorem . . . . . . . . . . . 522
B.2 Causal Conditions for ARMAModels . . . . . . . . . . . . . . 526
B.3 Large Sample Distribution of the AR(p)
Conditional Least Squares Estimators . . . . . . . . . . . . . . 528
B.4 TheWold Decomposition . . . . . . . . . . . . . . . . . . . . . 532
Appendix C: Spectral Domain Theory 534
C.1 Spectral Representation Theorem . . . . . . . . . . . . . . . . . 534
C.2 Large Sample Distribution of the DFT and
Smoothed Periodogram . . . . . . . . . . . . . . . . . . . . . . 539
C.3 The ComplexMultivariate Normal Distribution . . . . . . . . . 550
References 555
Index 569

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Introduction to Probability and Statistics for Engineers and Scientists

CONTENTS
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter 1 Introduction to Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Data Collection and Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Inferential Statistics and Probability Models . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Populations and Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 A Brief History of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 2 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Describing Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Frequency Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Relative Frequency Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.3 Grouped Data, Histograms, Ogives, and Stem and Leaf Plots . . . . . . . . . . . 14
2.3 Summarizing Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Sample Mean, Sample Median, and Sample Mode . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Sample Variance and Sample Standard Deviation . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Sample Percentiles and Box Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Chebyshev’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Normal Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Paired Data Sets and the Sample Correlation Coefficient . . . . . . . . . . . . . . 33
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Chapter 3 Elements of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Sample Space and Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Venn Diagrams and the Algebra of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 Axioms of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Sample Spaces Having Equally Likely Outcomes . . . . . . . . . . . . . . . . . . . . . 61
3.6 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Bayes’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.8 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Chapter 4 Random Variables and Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Types of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3 Jointly Distributed Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.1 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
*4.3.2 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5 Properties of the Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5.1 Expected Value of Sums of Random Variables . . . . . . . . . . . . . . . . . . . . . . . 115
4.6 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.7 Covariance and Variance of Sums of Random Variables . . . . . . . . . . . . . . 121
4.8 Moment Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.9 Chebyshev’s Inequality and the Weak Law of Large Numbers . . . . . . . . . 127
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Chapter 5 Special Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1 The Bernoulli and Binomial Random Variables . . . . . . . . . . . . . . . . . . . . . . 141
5.1.1 Computing the Binomial Distribution Function . . . . . . . . . . . . . . . . . . . . . 147
5.2 The Poisson Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.2.1 Computing the Poisson Distribution Function . . . . . . . . . . . . . . . . . . . . . . . 155
5.3 The Hypergeometric Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.4 The Uniform Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.5 Normal Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.6 Exponential Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
*5.6.1 The Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
*5.7 The Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.8 Distributions Arising from the Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.8.1 The Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
*5.8.1.1 The Relation Between Chi-Square and Gamma Random
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.8.2 The t-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.8.3 The F-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
*5.9 The Logistics Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Chapter 6 Distributions of Sampling Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.2 The Sample Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.3 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.3.1 Approximate Distribution of the Sample Mean . . . . . . . . . . . . . . . . . . . . . . 210
6.3.2 How Large a Sample is Needed? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.4 The Sample Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6.5 Sampling Distributions from a Normal Population . . . . . . . . . . . . . . . . . . . 214
6.5.1 Distribution of the Sample Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.5.2 Joint Distribution of X and S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.6 Sampling from a Finite Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Chapter 7 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.2 Maximum Likelihood Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
*7.2.1 Estimating Life Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.3 Interval Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
7.3.1 Confidence Interval for a Normal Mean When the Variance is
Unknown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.3.2 Confidence Intervals for the Variances of a Normal Distribution . . . . . . . . 251
7.4 Estimating the Difference in Means of Two Normal Populations . . . . . . 253
7.5 Approximate Confidence Interval for the Mean of a Bernoulli
Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
*7.6 Confidence Interval of the Mean of the Exponential Distribution . . . . . . 265
*7.7 Evaluating a Point Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
*7.8 The Bayes Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Chapter 8 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
8.2 Significance Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
8.3 Tests Concerning the Mean of a Normal Population . . . . . . . . . . . . . . . . . 293
8.3.1 Case of Known Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
8.3.2 Case of Unknown Variance: The t-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
8.4 Testing the Equality of Means of Two Normal Populations . . . . . . . . . . . 312
8.4.1 Case of Known Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
8.4.2 Case of Unknown Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
8.4.3 Case of Unknown and Unequal Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
8.4.4 The Paired t-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
8.5 Hypothesis Tests Concerning the Variance of a Normal Population . . . . 321
8.5.1 Testing for the Equality of Variances of Two Normal
Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
8.6 Hypothesis Tests in Bernoulli Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
8.6.1 Testing the Equality of Parameters in Two Bernoulli
Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
8.7 Tests Concerning the Mean of a Poisson Distribution . . . . . . . . . . . . . . . . 330
8.7.1 Testing the Relationship Between Two Poisson Parameters . . . . . . . . . . . . . 331
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
Chapter 9 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
9.2 Least Squares Estimators of the Regression Parameters . . . . . . . . . . . . . . . . 353
9.3 Distribution of the Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
9.4 Statistical Inferences about the Regression Parameters . . . . . . . . . . . . . . . . 361
9.4.1 Inferences Concerning β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
9.4.1.1 Regression to the Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
9.4.2 Inferences Concerning α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
9.4.3 Inferences Concerning the Mean Response α + βx0 . . . . . . . . . . . . . . . . . . 371
9.4.4 Prediction Interval of a Future Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
9.4.5 Summary of Distributional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
9.5 The Coefficient of Determination and the Sample Correlation
Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
9.6 Analysis of Residuals: Assessing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
9.7 Transforming to Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
9.8 Weighted Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
9.9 Polynomial Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
*9.10 Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
9.10.1 Predicting Future Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
9.11 Logistic Regression Models for Binary Output Data . . . . . . . . . . . . . . . . . 410
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
Chapter 10 Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
10.2 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
10.3 One-Way Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
10.3.1 Multiple Comparisons of Sample Means . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
10.3.2 One-Way Analysis of Variance with Unequal Sample Sizes . . . . . . . . . . . . 452
10.4 Two-Factor Analysis of Variance: Introduction and Parameter
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
10.5 Two-Factor Analysis of Variance: Testing Hypotheses . . . . . . . . . . . . . . . . 458
10.6 Two-Way Analysis of Variance with Interaction . . . . . . . . . . . . . . . . . . . . . 463
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
Chapter 11 Goodness of Fit Tests and Categorical Data Analysis . . . . . . . . 483
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
11.2 Goodness of Fit Tests When all Parameters are Specified . . . . . . . . . . . . . 484
11.2.1 Determining the Critical Region by Simulation . . . . . . . . . . . . . . . . . . . . . 490
11.3 Goodness of Fit Tests When Some Parameters are Unspecified . . . . . . . . 493
11.4 Tests of Independence in Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . 495
11.5 Tests of Independence in Contingency Tables Having Fixed
Marginal Totals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
*11.6 The Kolmogorov–Smirnov Goodness of Fit Test for Continuous
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
Chapter 12 Nonparametric Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
12.2 The Sign Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
12.3 The Signed Rank Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
12.4 The Two-Sample Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
12.4.1 The Classical Approximation and Simulation . . . . . . . . . . . . . . . . . . . . . . . 529
12.5 The Runs Test for Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
Chapter 13 Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
13.2 Control Charts for Average Values: The X -Control Chart . . . . . . . . . . . . 546
13.2.1 Case of Unknown μ and σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
13.3 S-Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
13.4 Control Charts for the Fraction Defective . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
13.5 Control Charts for Number of Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
13.6 Other Control Charts for Detecting Changes in the Population
Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
13.6.1 Moving-Average Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
13.6.2 Exponentially Weighted Moving-Average Control Charts . . . . . . . . . . . . . 565
13.6.3 Cumulative Sum Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Chapter 14* Life Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
14.2 Hazard Rate Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
14.3 The Exponential Distribution in Life Testing . . . . . . . . . . . . . . . . . . . . . . . . 584
14.3.1 Simultaneous Testing — Stopping at the rth Failure . . . . . . . . . . . . . . . . . 584
14.3.2 Sequential Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
14.3.3 Simultaneous Testing — Stopping by a Fixed Time . . . . . . . . . . . . . . . . . . 594
14.3.4 The Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
14.4 A Two-Sample Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
14.5 The Weibull Distribution in Life Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
14.5.1 Parameter Estimation by Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
Appendix of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

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Fundamentals of Probability and Statistics for Engineers

This textbook differs from others in the field in that it has been prepared very much with students and their needs in mind, having been classroom tested over many years. It is a true “learner’s book” made for students who require a deeper understanding of probability and statistics. It presents the fundamentals of the subject along with concepts of probabilistic modelling, and the process of model selection, verification and analysis. Furthermore, the inclusion of more than 100 examples and 200 exercises (carefully selected from a wide range of topics), along with a solutions manual for instructors, means that this text is of real value to students and lecturers across a range of engineering disciplines.
Key features:

Presents the fundamentals in probability and statistics along with relevant applications.
Explains the concept of probabilistic modelling and the process of model selection, verification and analysis.
Definitions and theorems are carefully stated and topics rigorously treated.
Includes a chapter on regression analysis.
Covers design of experiments.
Demonstrates practical problem solving throughout the book with numerous examples and exercises purposely selected from a variety of engineering fields.
Includes an accompanying online Solutions Manual for instructors containing complete step-by-step solutions to all problems.


Preface.
1. Introduction.

Part A: Probability and Random Variables.

2. Basic Probability Concepts.

3. Random Variables and Probability Distributions.

4. Expectations And Moments.

5. Functions of Random Variables.

6. Some Impotant Discrete Distributions.

7. Some Impotant Continuous Distributions.

Part B: Statistical Inference, Parameter Estimation, and Model Verification.

8. Observed Data and Graphical Representation.

9. Parameter Estimation.

10. Model Verification.

11. Linear Models and Linear Regression.

Appendix A: Tables.

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Probability and Statistics for Engineers and Scientists

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Elementary Statistics Using JMP

Contents
Acknowledgments ix
Part 1 The Basics 1
Chapter 1 Introduction 3
Purpose 4
Audience 4
What This Book Is and Isn’t 4
How This Book Is Organized 5
How to Use This Book 9
Using JMP or JMP SE 9
Chapter 2 Getting Started 11
JMP and Your Computer 12
Explaining JMP Terms 13
Starting JMP 14
Displaying a Simple Example 14
Getting Help 17
Exiting JMP 19
Chapter 3 Using Data Tables 21
Using Data Tables 22
What Is a Data Table? 22
Creating the Speeding Ticket Data Table 24
Opening an Existing JMP Data Table 29
Importing Data 29
Understanding Data Tables 36
Working with Columns 41
Working with Rows 48
Understanding Data Type and Modeling Type 50
Ordering Values 53
Adding Value Labels 56
Printing a Data Table 57
Sorting a Data Table 60
Summary 62
Exercises 70
Chapter 4 Summarizing Data 71
Checking Data for Errors 73
Using Distribution for a Continuous Variable 73
Using Distribution for Multiple Variables 89
Interacting with Distribution Results 94
Customizing Reports 100
Summaries 104
Exercises 111
Chapter 5 Graphing Data and Printing Results 113
Creating Custom Bar Charts 114
Creating Treemaps 122
Printing Results 127
Saving Results to Journals 129
Summaries 134
Exercises 138
Part 2 Statistical Background 139
Chapter 6 Understanding Fundamental Statistical
Concepts 141
Populations and Samples 142
The Normal Distribution 146
Parametric and Nonparametric Statistical Methods 150
Testing for Normality 150
Building a Hypothesis Test 164
Statistical and Practical Significance 166
Summaries 170
Exercises 172
Chapter 7 Estimating the Mean 173
Using One Number to Estimate the Mean 174
Effect of Sample Size 175
Effect of Population Variability 178
The Distribution of Sample Averages 179
Getting Confidence Intervals for the Mean 184
Summaries 189
Exercises 190
Part 3 Comparing Groups 191
Chapter 8 Comparing Two Groups 193
Deciding between Independent and Paired Groups 195
Summarizing Data from Two Independent Groups 196
Summarizing Data from Paired Groups 199
Building Hypothesis Tests to Compare Two Groups 205
Performing the Two-Sample t-test 207
Performing the Wilcoxon Rank Sum Test 220
Enhancing the Two-Sample Graph 224
Performing the Paired-Difference t-test 228
Performing the Wilcoxon Signed Rank Test 235
Summaries 239
Exercises 243
Special Topic: Paired Data in a Single Column 246
Chapter 9 Comparing More Than Two Groups 249
Summarizing Data from Multiple Groups 251
Building Hypothesis Tests to Compare More Than Two Groups 257
Performing a One-way Analysis of Variance 259
Analysis of Variance with Unequal Variances 266
Performing a Kruskal-Wallis Test 270
Enhancing JMP Graphs 273
Multiple Comparison Procedures 274
Summarizing with an Example 293
Summary 299
Exercises 302
Part 4 Fitting Lines to Data 305
Chapter 10 Correlation and Regression 307
Summarizing Multiple Continuous Variables 309
Calculating Correlation Coefficients 316
Performing Straight-Line Regression 320
Fitting a Straight Line Using JMP 324
Summarizing Straight-Line Regression 335
Fitting Curves 336
Regression with Two or More Independent Variables 346
Summaries 353
Exercises 357
Chapter 11 Basic Regression Diagnostics 359
Concepts in Plotting Residuals 360
Creating Residuals Plots for the Energy Data 363
Creating Residuals Plots for the Engine Data 375
Using the Lack Of Fit Report 384
Testing the Regression Assumption for Errors 388
Summaries 392
Exercises 396
Special Topic: Leverage Plots 397
Part 5 Data in Summary Tables 401
Chapter 12 Creating and Analyzing Contingency Tables 403
Defining Contingency Tables 404
Summarizing Raw Data in Tables 405
Creating a JMP Contingency Table from an Existing
Summary Table 411
Creating Contingency Tables for Several Variables 414
Performing Tests for Independence 419
Measures of Association with Ordinal Variables 424
Summaries 429
Exercises 432
Special Topic: Statistical Summary Tables 434
Appendix 1 Further Reading 437
Statistics References 437
JMP Documentation 440
Index 441

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