# Introduction to Probability and Statistics for Engineers and Scientists, 6th Edition PDF by Sheldon M Ross

## Introduction to Probability and Statistics for Engineers and Scientists, Sixth Edition

By Sheldon M. Ross

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………………………………………… 4

Problems ………………………………………………………………… 7

CHAPTER 2 Descriptive statistics………………………………………………………. 11

2.1 Introduction …………………………………………………………… 11

2.2 Describing data sets ………………………………………………. 12

2.2.1 Frequency tables and graphs ……………………….. 12

2.2.2 Relative frequency tables and graphs……………. 14

2.2.3 Grouped data, histograms, ogives, and

stem and leaf plots………………………………………. 16

2.3 Summarizing data sets…………………………………………… 19

2.3.1 Sample mean, sample median, and sample

mode…………………………………………………………… 19

2.3.2 Sample variance and sample standard

deviation……………………………………………………… 24

2.3.3 Sample percentiles and box plots …………………. 26

2.4 Chebyshev’s inequality …………………………………………… 29

2.5 Normal data sets …………………………………………………… 33

2.6 Paired data sets and the sample

correlation coefficient…………………………………………….. 36

2.7 The Lorenz curve and Gini index……………………………… 43

2.8 Using R …………………………………………………………………. 48

Problems ………………………………………………………………. 52

CHAPTER 3 Elements of probability…………………………………………………… 63

3.1 Introduction …………………………………………………………… 63

3.2 Sample space and events……………………………………….. 64

3.3 Venn diagrams and the algebra of events………………… 66

3.4 Axioms of probability ……………………………………………… 67

3.5 Sample spaces having equally likely outcomes………… 70

3.6 Conditional probability……………………………………………. 75

3.7 Bayes’ formula ………………………………………………………. 79

3.8 Independent events ……………………………………………….. 86

Problems ………………………………………………………………. 89

CHAPTER 4 Random variables and expectation………………………………….. 99

4.1 Random variables ………………………………………………….. 99

4.2 Types of random variables ……………………………………. 102

4.3 Jointly distributed random variables……………………… 105

4.3.1 Independent random variables……………………. 111

4.3.2 Conditional distributions…………………………….. 114

4.4 Expectation………………………………………………………….. 117

4.5 Properties of the expected value …………………………… 121

4.5.1 Expected value of sums of random variables.. 124

4.6 Variance………………………………………………………………. 128

4.7 Covariance and variance of sums of

random variables …………………………………………………. 132

4.8 Moment generating functions……………………………….. 138

4.9 Chebyshev’s inequality and the weak law of large

numbers ……………………………………………………………… 139

Problems …………………………………………………………….. 142

CHAPTER 5 Special random variables……………………………………………… 151

5.1 The Bernoulli and binomial random variables ……….. 151

5.1.1 Using R to calculate binomial probabilities….. 157

5.2 The Poisson random variable………………………………… 158

5.2.1 Using R to calculate Poisson probabilities …… 166

5.3 The hypergeometric random variable ……………………. 167

5.4 The uniform random variable ……………………………….. 171

5.5 Normal random variables …………………………………….. 179

5.6 Exponential random variables ………………………………. 190

5.6.1 The Poisson process ………………………………….. 193

5.6.2 The Pareto distribution ………………………………. 196

5.7 The gamma distribution ……………………………………….. 199

5.8 Distributions arising from the normal …………………… 201

5.8.1 The chi-square distribution ………………………… 201

5.8.2 The t -distribution ………………………………………. 206

5.8.3 The F -distribution ……………………………………… 208

5.9 The logistics distribution ………………………………………. 209

5.10 Distributions in R …………………………………………………. 210

Problems …………………………………………………………….. 212

CHAPTER 6 Distributions of sampling statistics ……………………………….. 221

6.1 Introduction …………………………………………………………. 221

6.2 The sample mean ………………………………………………… 222

6.3 The central limit theorem …………………………………….. 224

6.3.1 Approximate distribution of the sample mean 227

6.3.2 How large a sample is needed?…………………… 230

6.4 The sample variance…………………………………………….. 230

6.5 Sampling distributions from a normal population ….. 231

6.5.1 Distribution of the sample mean…………………. 232

6.5.2 Joint distribution of X and S2 ……………………… 232

6.6 Sampling from a finite population …………………………. 234

Problems …………………………………………………………….. 238

CHAPTER 7 Parameter estimation…………………………………………………… 245

7.1 Introduction …………………………………………………………. 245

7.2 Maximum likelihood estimators ……………………………. 246

7.2.1 Estimating life distributions ……………………….. 255

7.3 Interval estimates ………………………………………………… 257

7.3.1 Confidence interval for a normal mean when

the variance is unknown …………………………….. 262

7.3.2 Prediction intervals ……………………………………. 268

7.3.3 Confidence intervals for the variance of a

normal distribution ……………………………………. 269

7.4 Estimating the difference in means of two normal

populations………………………………………………………….. 270

7.5 Approximate confidence interval for the mean of a

Bernoulli random variable ……………………………………. 275

7.6 Confidence interval of the mean of the exponential

distribution …………………………………………………………. 280

7.7 Evaluating a point estimator …………………………………. 281

7.8 The Bayes estimator…………………………………………….. 287

Problems …………………………………………………………….. 292

CHAPTER 8 Hypothesis testing ……………………………………………………….. 305

8.1 Introduction …………………………………………………………. 305

8.2 Significance levels ……………………………………………….. 306

8.3 Tests concerning the mean of a normal population… 307

8.3.1 Case of known variance ……………………………… 307

8.3.2 Case of unknown variance: the t -test………….. 319

8.4 Testing the equality of means of two normal

populations………………………………………………………….. 326

8.4.1 Case of known variances ……………………………. 326

8.4.2 Case of unknown variances ………………………… 328

8.4.3 Case of unknown and unequal variances …….. 333

8.4.4 The paired t -test………………………………………… 333

8.5 Hypothesis tests concerning the variance of a

normal population ……………………………………………….. 336

8.5.1 Testing for the equality of variances of two

normal populations ……………………………………. 337

8.6 Hypothesis tests in Bernoulli populations ……………… 339

8.6.1 Testing the equality of parameters in two

Bernoulli populations…………………………………. 342

8.7 Tests concerning the mean of a Poisson distribution 345

8.7.1 Testing the relationship between two

Poisson parameters …………………………………… 346

Problems …………………………………………………………….. 348

CHAPTER 9 Regression…………………………………………………………………… 365

9.1 Introduction …………………………………………………………. 365

9.2 Least squares estimators of the

regression parameters…………………………………………. 367

9.3 Distribution of the estimators……………………………….. 371

regression parameters…………………………………………. 377

9.4.1 Inferences concerning β …………………………….. 377

9.4.2 Inferences concerning α …………………………….. 386

9.4.3 Inferences concerning the mean response

α + βx0 ……………………………………………………… 386

9.4.4 Prediction interval of a future response ………. 389

9.4.5 Summary of distributional results ………………. 392

9.5 The coefficient of determination and the sample

correlation coefficient…………………………………………… 392

9.6 Analysis of residuals: assessing the model ……………. 395

9.7 Transforming to linearity………………………………………. 396

9.8 Weighted least squares ………………………………………… 400

9.9 Polynomial regression………………………………………….. 406

9.10 Multiple linear regression …………………………………….. 410

9.10.1 Predicting future responses ……………………….. 420

9.10.2 Dummy variables for categorical data…………. 424

9.11 Logistic regression models for binary output data….. 425

Problems …………………………………………………………….. 429

CHAPTER 10 Analysis of variance ……………………………………………………… 453

10.1 Introduction …………………………………………………………. 453

10.2 An overview …………………………………………………………. 454

10.3 One-way analysis of variance………………………………… 456

10.3.1 Using R to do the computations ………………….. 463

10.3.2 Multiple comparisons of sample means………. 466

10.3.3 One-way analysis of variance with unequal

sample sizes ……………………………………………… 468

10.4 Two-factor analysis of variance: introduction and

parameter estimation…………………………………………… 470

10.5 Two-factor analysis of variance: testing hypotheses.. 474

10.6 Two-way analysis of variance with interaction ……….. 479

Problems …………………………………………………………….. 487

CHAPTER 11 Goodness of fit tests and categorical data analysis…………. 499

11.1 Introduction …………………………………………………………. 499

11.2 Goodness of fit tests when all parameters are

specified ……………………………………………………………… 500

11.2.1 Determining the critical region by simulation. 506

11.3 Goodness of fit tests when some parameters are

unspecified ………………………………………………………….. 508

11.4 Tests of independence in contingency tables………….. 510

11.5 Tests of independence in contingency tables having

fixed marginal totals…………………………………………….. 514

11.6 The Kolmogorov–Smirnov goodness of fit test for

continuous data……………………………………………………. 517

Problems …………………………………………………………….. 522

CHAPTER 12 Nonparametric hypothesis tests……………………………………. 529

12.1 Introduction …………………………………………………………. 529

12.2 The sign test………………………………………………………… 529

12.3 The signed rank test …………………………………………….. 533

12.4 The two-sample problem……………………………………… 538

12.4.1 Testing the equality of multiple probability distributions………………………………………………. 541

12.5 The runs test for randomness ………………………………. 544

Problems …………………………………………………………….. 547

CHAPTER 13 Quality control ……………………………………………………………… 555

13.1 Introduction …………………………………………………………. 555

13.2 Control charts for average values: the x control

chart……………………………………………………………………. 556

13.2.1 Case of unknown μ and σ …………………………… 559

13.3 S-control charts …………………………………………………… 564

13.4 Control charts for the fraction defective ………………… 567

13.5 Control charts for number of defects…………………….. 569

13.6 Other control charts for detecting changes in the population mean ………………….. 573

13.6.1 Moving-average control charts …………………… 573

13.6.2 Exponentially weighted moving-average control charts …………………………… 576

13.6.3 Cumulative sum control charts…………………… 581

Problems …………………………………………………………….. 583

CHAPTER 14 Life testing∗ …………………………………………………………………. 591

14.1 Introduction …………………………………………………………. 591

14.2 Hazard rate functions …………………………………………… 591

14.3 The exponential distribution in life testing……………… 594

14.3.1 Simultaneous testing — stopping at the rth

failure ……………………………………………………….. 594

14.3.2 Sequential testing ……………………………………… 599

14.3.3 Simultaneous testing — stopping by a fixed

time ………………………………………………………….. 603

14.3.4 The Bayesian approach………………………………. 606

14.4 A two-sample problem…………………………………………. 607

14.5 The Weibull distribution in life testing……………………. 609

14.5.1 Parameter estimation by least squares……….. 611

Problems …………………………………………………………….. 613

CHAPTER 15 Simulation, bootstrap statistical methods, and

permutation tests…………………………………………………………. 619

15.1 Introduction …………………………………………………………. 619

15.2 Random numbers ………………………………………………… 619

15.2.1 The Monte Carlo simulation approach…………. 622

15.3 The bootstrap method ………………………………………….. 623

15.4 Permutation tests ………………………………………………… 631

15.4.1 Normal approximations in permutation tests . 634

15.4.2 Two-sample permutation tests …………………… 637

15.5 Generating discrete random variables…………………… 639

15.6 Generating continuous random variables………………. 641

15.6.1 Generating a normal random variable…………. 643

15.7 Determining the number of simulation runs

in a Monte Carlo study………………………………………….. 644

Problems …………………………………………………………….. 645

CHAPTER 16 Machine learning and big data………………………………………. 649

16.1 Introduction …………………………………………………………. 649

16.2 Late flight probabilities ………………………………………… 650

16.3 The naive Bayes approach…………………………………….. 651

16.3.1 A variation of naive Bayes approach ……………. 654

16.4 Distance-based estimators. The k-nearest neighbors rule…………………. 657

16.4.1 A distance-weighted method………………………. 658

16.4.2 Component-weighted distances………………….. 659

16.5 Assessing the approaches…………………………………….. 660

16.6 When characterizing vectors are quantitative ………… 662

16.6.1 Nearest neighbor rules………………………………. 662

16.6.2 Logistics regression …………………………………… 663

16.7 Choosing the best probability: a bandit problem…….. 664

Problems …………………………………………………………….. 666

APPENDIX OF TABLES ……………………………………………………………………….. 669

INDEX ……………………………………………………………………………………………….. 673

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