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Probability with STEM Applications. Edition No. 3

  • Book

  • 640 Pages
  • March 2021
  • John Wiley and Sons Ltd
  • ID: 5842025

Probability with STEM Applications, Third Edition, is an accessible and well-balanced introduction to post-calculus applied probability. Integrating foundational mathematical theory and the application of probability in the real world, this leading textbook engages students with unique problem scenarios and more than 1100 exercises of varying levels of difficulty. The text uses a hands-on, software-oriented approach to the subject of probability. MATLAB and R examples and exercises - complemented by computer code that enables students to create their own simulations - demonstrate the importance of software to solve problems that cannot be obtained analytically.

Revised and updated throughout, the textbook covers basic properties of probability, random variables and their probability distributions, a brief introduction to statistical inference, Markov chains, stochastic processes, and signal processing. This new edition is the perfect text for a one-semester course and contains enough additional material for an entire academic year. The blending of theory and application will appeal not only to mathematics and statistics majors but also to engineering students, and quantitative business and social science majors.

New to this Edition:

  • Offered as a traditional textbook and in enhanced ePub format, containing problems with show/hide solutions and interactive applets and illustrations
  • Revised and expanded chapters on conditional probability and independence, families of continuous distributions, and Markov chains
  • New problems and updated problem sets throughout

Features:

  • Introduces basic theoretical knowledge in the first seven chapters, serving as a self-contained textbook of roughly 650 problems
  • Provides numerous up-to-date examples and problems in R and MATLAB
  • Discusses examples from recent journal articles, classic problems, and various practical applications
  • Includes a chapter specifically designed for electrical and computer engineers, suitable for a one-term class on random signals and noise
  • Contains appendices of statistical tables, background mathematics, and important probability distributions

Table of Contents

Preface xv

Introduction 1

Why Study Probability? 1

Software Use in Probability 2

Modern Application of Classic Probability Problems 2

Applications to Business 3

Applications to the Life Sciences 4

Applications to Engineering and Operations Research 4

Applications to Finance 6

Probability in Everyday Life 7

1 Introduction to Probability 13

Introduction 13

1.1 Sample Spaces and Events 13

The Sample Space of an Experiment 13

Events 15

Some Relations from Set Theory 16

Exercises Section 1.1 (1-12) 18

1.2 Axioms Interpretations and Properties of Probability 19

Interpreting Probability 21

More Probability Properties 23

Contingency Tables 25

Determining Probabilities Systematically 26

Equally Likely Outcomes 27

Exercises Section 1.2 (13-30) 28

1.3 Counting Methods 30

The Fundamental Counting Principle 31

Tree Diagrams 32

Permutations 33

Combinations 34

Partitions 38

Exercises Section 1.3 (31-50) 39

Supplementary Exercises (51-62) 42

2 Conditional Probability and Independence 45

Introduction 45

2.1 Conditional Probability 45

The Definition of Conditional Probability 46

The Multiplication Rule for P(A ∩ B) 49

2.2 The Law of Total Probability and Bayes’ Theorem 52

The Law of Total Probability 52

Bayes’ Theorem 55

Exercises Section 2.2 (17-32) 59

2.3 Independence 61

The Multiplication Rule for Independent Events 63

Independence of More Than Two Events 65

Exercises Section 2.3 (33-54) 66

2.4 Simulation of Random Events 69

The Backbone of Simulation: Random Number Generators 70

Precision of Simulation 73

Exercises Section 2.4 (55-74) 74

Supplementary Exercises (75-100) 77

3 Discrete Probability Distributions:general Properties 82

Introduction 82

3.1 Random Variables 82

Two Types of Random Variables 84

Exercises Section 3.1 (1-10) 85

3.2 Probability Distributions for Discrete Random Variables 86

Another View of Probability Mass Functions 89

Exercises Section 3.2 (11-21) 90

3.3 The Cumulative Distribution Function 91

Exercises Section 3.3 (22-30) 95

3.4 Expected Value and Standard Deviation 96

The Expected Value of X 97

The Expected Value of a Function 99

The Variance and Standard Deviation of X 102

Properties of Variance 104

Exercises Section 3.4 (31-50) 105

3.5 Moments and Moment Generating Functions 108

The Moment Generating Function 109

Obtaining Moments from the MGF 111

Exercises Section 3.5 (51-64) 113

3.6 Simulation of Discrete Random Variables 114

Simulations Implemented in R and Matlab 117

Simulation Mean Standard Deviation and Precision 117

Exercises Section 3.6 (65-74) 119

Supplementary Exercises (75-84) 120

4 Families of Discrete Distributions 122

Introduction 122

4.1 Parameters and Families of Distributions 122

Exercises Section 4.1 (1-6) 124

4.2 The Binomial Distribution 125

The Binomial Random Variable and Distribution 127

Computing Binomial Probabilities 129

The Mean Variance and Moment Generating Function 130

Binomial Calculations with Software 132

Exercises Section 4.2 (7-34) 132

4.3 The Poisson Distribution 136

The Poisson Distribution as a Limit 137

The Mean Variance and Moment Generating Function 139

The Poisson Process 140

Poisson Calculations with Software 141

Exercises Section 4.3 (35-54) 142

4.4 The Hypergeometric Distribution 145

Mean and Variance 148

Hypergeometric Calculations with Software 149

Exercises Section 4.4 (55-64) 149

4.5 The Negative Binomial and Geometric Distributions 151

The Geometric Distribution 152

Mean Variance and Moment Generating Function 152

Alternative Definitions of the Negative Binomial Distribution 153

Negative Binomial Calculations with Software 154

Exercises Section 4.5 (65-78) 154

Supplementary Exercises (79-100) 156

5 Continuous Probability Distributions:general Properties 160

Introduction 160

5.1 Continuous Random Variables and Probability Density Functions 160

Probability Distributions for Continuous Variables 161

Exercises Section 5.1 (1-8) 165

5.2 The Cumulative Distribution Function and Percentiles 166

Using F(x) to Compute Probabilities 168

Obtaining f(x) fromF(x) 169

Percentiles of a Continuous Distribution 169

Exercises Section 5.2 (9-18) 171

5.3 Expected Values Variance and Moment Generating Functions 173

Expected Values 173

Variance and Standard Deviation 175

Properties of Expectation and Variance 176

Moment Generating Functions 177

Exercises Section 5.3 (19-38) 179

5.4 Transformation of a Random Variable 181

Exercises Section 5.4 (39-54) 185

5.5 Simulation of Continuous Random Variables 186

The Inverse CDF Method 186

The Accept-Reject Method 189

Precision of Simulation Results 191

Exercises Section 5.5 (55-63) 191

Supplementary Exercises (64-76) 193

6 Families of Continuous Distributions 196

Introduction 196

6.1 The Normal (Gaussian) Distribution 196

The Standard Normal Distribution 197

Arbitrary Normal Distributions 199

The Moment Generating Function 203

Normal Distribution Calculations with Software 204

Exercises Section 6.1 (1-27) 205

6.2 Normal Approximation of Discrete Distributions 208

Approximating the Binomial Distribution 209

Exercises Section 6.2 (28-36) 211

6.3 The Exponential and Gamma Distributions 212

The Exponential Distribution 212

The Gamma Distribution 214

The Gamma and Exponential MGFs 217

Gamma and Exponential Calculations with Software 218

Exercises Section 6.3 (37-50) 218

6.4 Other Continuous Distributions 220

The Weibull Distribution 220

The Lognormal Distribution 222

The Beta Distribution 224

Exercises Section 6.4 (51-66) 226

6.5 Probability Plots 228

Sample Percentiles 228

A Probability Plot 229

Departures from Normality 232

Beyond Normality 234

Probability Plots in Matlab and R 236

Exercises Section 6.5 (67-76) 237

Supplementary Exercises (77-96) 238

7 Joint Probability Distributions 242

Introduction 242

7.1 Joint Distributions for Discrete Random Variables 242

The Joint Probability Mass Function for Two Discrete Random Variables 242

Marginal Probability Mass Functions 244

Independent Random Variables 245

More Than Two Random Variables 246

Exercises Section 7.1 (1-12) 248

7.2 Joint Distributions for Continuous Random Variables 250

The Joint Probability Density Function for Two Continuous Random Variables 250

Marginal Probability Density Functions 252

Independence of Continuous Random Variables 254

More Than Two Random Variables 255

Exercises Section 7.2 (13-22) 257

7.3 Expected Values Covariance and Correlation 258

Properties of Expected Value 260

Covariance 261

Correlation 263

Correlation Versus Causation 265

Exercises Section 7.3 (23-42) 266

7.4 Properties of Linear Combinations 267

Expected Value and Variance of a Linear Combination 268

The PDF of a Sum 271

Moment Generating Functions of Linear Combinations 273

Exercises Section 7.4 (43-65) 275

7.5 The Central Limit Theorem and the Law of Large Numbers 278

Random Samples 278

The Central Limit Theorem 282

A More General Central Limit Theorem 286

Other Applications of the Central Limit Theorem 287

The Law of Large Numbers 288

Proof of the Central Limit Theorem 290

Exercises Section 7.5 (66-82) 290

7.6 Simulation of Joint Probability Distributions 293

Simulating Values from a Joint PMF 293

Simulating Values from a Joint PDF 295

Exercises Section 7.6 (83-90) 297

Supplementary Exercises (91-124) 298

8 Joint Probability Distributions:additional Topics 304

Introduction 304

8.1 Conditional Distributions and Expectation 304

Conditional Distributions and Independence 306

Conditional Expectation and Variance 307

The Laws of Total Expectation and Variance 308

Exercises Section 8.1 (1-18) 313

8.2 The Bivariate Normal Distribution 315

Conditional Distributions of X and Y 317

Regression to the Mean 318

The Multivariate Normal Distribution 319

Bivariate Normal Calculations with Software 319

Exercises Section 8.2 (19-30) 320

8.3 Transformations of Jointly Distributed Random Variables 321

The Joint Distribution of Two New Random Variables 322

The Distribution of a Single New RV 323

The Joint Distribution of More Than Two New Variables 325

Exercises Section 8.3 (31-38) 326

8.4 Reliability 327

The Reliability Function 327

Series and Parallel System Designs 329

Mean Time to Failure 331

The Hazard Function 332

Exercises Section 8.4 (39-50) 335

8.5 Order Statistics 337

The Distributions of Yn and Y1  337

The Distribution of the ith Order Statistic 339

The Joint Distribution of All n Order Statistics 340

Exercises Section 8.5 (51-60) 342

8.6 Further Simulation Tools for Jointly Distributed Random Variables 343

The Conditional Distribution Method of Simulation 343

Simulating a Bivariate Normal Distribution 344

Simulation Methods for Reliability 346

Exercises Section 8.6 (61-68) 347

Supplementary Exercises (69-82) 348

9 the Basics of Statistical Inference 351

Introduction 351

9.1 Point Estimation 351

Estimates and Estimators 352

Assessing Estimators: Accuracy and Precision 354

Exercises Section 9.1 (1-18) 357

9.2 Maximum Likelihood Estimation 360

Some Properties of MLEs 366

Exercises Section 9.2 (19-30) 367

9.3 Statistical Intervals 368

Constructing a Confidence Interval 369

Confidence Intervals for a Population Proportion 369

Confidence Intervals for a Population Mean 371

Further Comments on Statistical Intervals 375

Confidence Intervals with Software 375

Exercises Section 9.3 (31-48) 376

9.4 Hypothesis Tests 379

Hypotheses and Test Procedures 380

Hypothesis Testing for a Population Mean 381

Errors in Hypothesis Testing and the Power of a Test 385

Hypothesis Testing for a Population Proportion 388

Software for Hypothesis Test Calculations 389

Exercises Section 9.4 (49-71) 391

9.5 Bayesian Estimation 393

The Posterior Distribution of a Parameter 394

Inferences from the Posterior Distribution 397

Further Comments on Bayesian Inference 398

Exercises Section 9.5 (72-80) 399

9.6 Simulation-Based Inference 400

The Bootstrap Method 400

Interval Estimation Using the Bootstrap 402

Hypothesis Tests Using the Bootstrap 404

More on Simulation-Based Inference 405

Exercises Section 9.6 (81-90) 405

Supplementary Exercises (91-116) 407

10 Markov Chains 411

Introduction 411

10.1 Terminology and Basic Properties 411

The Markov Property 413

Exercises Section 10.1 (1-10) 416

10.2 The Transition Matrix and the Chapman-Kolmogorov Equations 418

The Transition Matrix 418

Computation of Multistep Transition Probabilities 419

Exercises Section 10.2 (11-22) 423

10.3 Specifying an Initial Distribution 426

A Fixed Initial State 428

Exercises Section 10.3 (23-30) 429

10.4 Regular Markov Chains and the Steady-State Theorem 430

Regular Chains 431

The Steady-State Theorem 432

Interpreting the Steady-State Distribution 433

Efficient Computation of Steady-State Probabilities 435

Irreducible and Periodic Chains 437

Exercises Section 10.4 (31-43) 438

10.5 Markov Chains with Absorbing States 440

Time to Absorption 441

Mean Time to Absorption 444

Mean First Passage Times 448

Probabilities of Eventual Absorption 449

Exercises Section 10.5 (44-58) 451

10.6 Simulation of Markov Chains 453

Exercises Section 10.6 (59-66) 459

Supplementary Exercises (67-82) 461

11 Random Processes 465

Introduction 465

11.1 Types of Random Processes 465

Classification of Processes 468

Random Processes and Their Associated Random Variables 469

Exercises Section 11.1 (1-10) 470

11.2 Properties of the Ensemble: Mean and Autocorrelation Functions 471

Mean and Variance Functions 471

Autocovariance and Autocorrelation Functions 475

The Joint Distribution of Two Random Processes 477

Exercises Section 11.2 (11-24) 478

11.3 Stationary and Wide-Sense Stationary Processes 479

Properties of WSS Processes 483

Ergodic Processes 486

Exercises Section 11.3 (25-40) 488

11.4 Discrete-Time Random Processes 489

Special Discrete Sequences 491

Exercises Section 11.4 (41-52) 493

Supplementary Exercises (53-64) 494

12 Families of Random Processes 497

Introduction 497

12.1 Poisson Processes 497

Relation to Exponential and Gamma Distributions 499

Combining and Decomposing Poisson Processes 502

Alternative Definition of a Poisson Process 504

Nonhomogeneous Poisson Processes 505

The Poisson Telegraphic Process 506

Exercises Section 12.1 (1-18) 507

12.2 Gaussian Processes 509

Brownian Motion 510

Brownian Motion as a Limit 512

Further Properties of Brownian Motion 512

Variations on Brownian Motion 514

Exercises Section 12.2 (19-28) 515

12.3 Continuous-Time Markov Chains 516

Infinitesimal Parameters and Instantaneous Transition Rates 518

Sojourn Times and Transitions 520

Long-Run Behavior of Continuous-Time Markov Chains 523

Explicit Form of the Transition Matrix 526

Exercises Section 12.3 (29-40) 527

Supplementary Exercises (41-51) 529

13 Introduction to Signal Processing 532

Introduction 532

13.1 Power Spectral Density 532

Expected Power and the Power Spectral Density 532

Properties of the Power Spectral Density 535

Power in a Frequency Band 538

White Noise Processes 539

Cross-Power Spectral Density for Two Processes 541

Exercises Section 13.1 (1-21) 542

13.2 Random Processes and LTI Systems 544

Properties of the LTI System Output 545

Ideal Filters 548

Signal Plus Noise 551

Exercises Section 13.2 (22-38) 554

13.3 Discrete-Time Signal Processing 556

Random Sequences and LTI Systems 558

Sampling Random Sequences 560

Exercises Section 13.3 (39-50) 562

A Statistical Tables A- 1

A 1 Binomial CDF A- 1

A 2 Poisson CDF A- 4

A 3 Standard Normal CDF A- 5

A 4 Incomplete Gamma Function A- 7

A 5 Critical Values for t Distributions A- 7

A 6 Tail Areas of t Distributions A- 9

B Background Mathematics A- 13

B 1 Trigonometric Identities A- 13

B 2 Special Engineering Functions A- 13

B 3 o(h) Notation A- 14

B 4 The Delta Function A- 14

B 5 Fourier Transforms A- 15

B 6 Discrete-Time Fourier Transforms A- 16

C Important Probability Distributions A- 18

C 1 Discrete Distributions A- 18

C 2 Continuous Distributions A- 20

C 3 Matlab and R Commands A- 23

Bibliography B- 1

Answers to Odd-numbered Exercises S- 1

Index I- 1 

Authors

Matthew A. Carlton California Polytechnic State University. Jay L. Devore California Polytechnic State University.