# Probability

## With Applications and R

2. Edition August 2021

544 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-69238-6**

Discover the latest edition of a practical introduction to the theory of probability, complete with R code samples

In the newly revised Second Edition of Probability: With Applications and R, distinguished researchers Drs. Robert Dobrow and Amy Wagaman deliver a thorough introduction to the foundations of probability theory. The book includes a host of chapter exercises, examples in R with included code, and well-explained solutions. With new and improved discussions on reproducibility for random numbers and how to set seeds in R, and organizational changes, the new edition will be of use to anyone taking their first probability course within a mathematics, statistics, engineering, or data science program.

New exercises and supplemental materials support more engagement with R, and include new code samples to accompany examples in a variety of chapters and sections that didn't include them in the first edition.

The new edition also includes for the first time:

* A thorough discussion of reproducibility in the context of generating random numbers

* Revised sections and exercises on conditioning, and a renewed description of specifying PMFs and PDFs

* Substantial organizational changes to improve the flow of the material

* Additional descriptions and supplemental examples to the bivariate sections to assist students with a limited understanding of calculus

Perfect for upper-level undergraduate students in a first course on probability theory, Probability: With Applications and R is also ideal for researchers seeking to learn probability from the ground up or those self-studying probability for the purpose of taking advanced coursework or preparing for actuarial exams.

Acknowledgments xvii

Introduction xix

1 First Principles 1

1.1 Random Experiment, Sample Space, Event 1

1.2 What Is a Probability? 3

1.3 Probability Function 4

1.4 Properties of Probabilities 7

1.5 Equally Likely Outcomes 11

1.6 Counting I 12

1.6.1 Permutations 13

1.7 Counting II 16

1.7.1 Combinations and Binomial Coefficients 17

1.8 Problem-Solving Strategies: Complements and

Inclusion-Exclusion 26

1.9 A First Look at Simulation 29

1.10 Summary 34

Exercises 36

2 Conditional Probability and Independence 45

2.1 Conditional Probability 45

2.2 New Information Changes the Sample Space 50

2.3 Finding P (A and B) 51

2.3.1 Birthday Problem 56

2.4 Conditioning and the Law of Total Probability 60

2.5 Bayes Formula and Inverting a Conditional Probability 67

2.6 Independence and Dependence 72

2.7 Product Spaces 80

2.8 Summary 82

Exercises 83

3 Introduction to Discrete Random Variables 93

3.1 Random Variables 93

3.2 Independent Random Variables 97

3.3 Bernoulli Sequences 99

3.4 Binomial Distribution 101

3.5 Poisson Distribution 108

3.5.1 Poisson Approximation of Binomial Distribution 113

3.5.2 Poisson as Limit of Binomial Probabilitie; 115

3.6 Summary 116

Exercises 118

4 Expectation and More with Discrete Random Variables 125

4.1 Expectation 127

4.2 Functions of Random Variables 130

4.3 Joint Distributions 134

4.4 Independent Random Variables 139

4.4.1 Sums of Independent Random Variables 142

4.5 Linearity of Expectation 144

4.6 Variance and Standard Deviation 149

4.7 Covariance and Correlation 158

4.8 Conditional Distribution 165

4.8.1 Introduction to Conditional Expectation 168

4.9 Properties of Covariance and Correlation 171

4.10 Expectation of a Function of a Random Variable 173

4.11 Summary 174

Exercises 176

5 More Discrete Distributions and Their Relationships 185

5.1 Geometric Distribution 185

5.1.1 Memorylessness 188

5.1.2 Coupon Collecting and Tiger Counting 189

5.2 Moment-Generating Functions 193

5.3 Negative Binomial--Up from the Geometric 196

5.4 Hypergeometric--Sampling Without Replacement 202

5.5 From Binomial to Multinomial 207

5.6 Benford's Law 213

5.7 Summary 216

Exercises 218

6 Continuous Probability 227

6.1 Probability Density Function 229

6.2 Cumulative Distribution Function 233

6.3 Expectation and Variance 237

6.4 Uniform Distribution 239

6.5 Exponential Distribution 242

6.5.1 Memorylessness 243

6.6 Joint Distributions 247

6.7 Independence 256

6.7.1 Accept-Reject Method 258

6.8 Covariance, Correlation 262

6.9 Summary 264

Exercises 266

7 Continuous Distributions 273

7.1 Normal Distribution 273

7.1.1 Standard Normal Distribution 276

7.1.2 Normal Approximation of Binomial Distribution 278

7.1.3 Quantiles 282

7.1.4 Sums of Independent Normals 285

7.2 Gamma Distribution 288

7.2.1 Probability as a Technique of Integration 292

7.3 Poisson Process 294

7.4 Beta Distribution 302

7.5 Pareto Distribution 305

7.6 Summary 308

Exercises 311

8 Densities of Functions of Random Variables 319

8.1 Densities via CDFs 320

8.1.1 Simulating a Continuous Random Variable 326

8.1.2 Method of Transformations 329

8.2 Maximums, Minimums, and Order Statistics 330

8.3 Convolution 335

8.4 Geometric Probability 338

8.5 Transformations of Two Random Variables 344

8.6 Summary 348

Exercises 349

9 Conditional Distribution, Expectation, and Variance 357

Introduction 357

9.1 Conditional Distributions 358

9.2 Discrete and Continuous: Mixing it Up 364

9.3 Conditional Expectation 369

9.3.1 From Function to Random Variable 371

9.3.2 Random Sum of Random Variables 378

9.4 Computing Probabilities by Conditioning 378

9.5 Conditional Variance 382

9.6 Bivariate Normal Distribution 387

9.7 Summary 396

Exercises 398

10 LIMITS 407

10.1 Weak Law of Large Numbers 409

10.1.1 Markov and Chebyshev Inequalities 411

10.2 Strong Law of Large Numbers 415

10.3 Method of Moments 421

10.4 Monte Carlo Integration 424

10.5 Central Limit Theorem 428

10.5.1 Central Limit Theorem and Monte Carlo 436

10.6 A Proof of the Central Limit Theorem 437

10.7 Summary 439

Exercises 440

11 Beyond Random Walks and Markov Chains 447

11.1 Random Walk on Graphs 447

11.1.1 Long-Term Behavior 451

11.2 Random Walks on Weighted Graphs and Markov Chains 455

11.2.1 Stationary Distribution 458

11.3 From Markov Chain to Markov Chain Monte Carlo 462

11.4 Summary 474

Exercises 476

Appendix A Probability Distributions in R 481

Appendix B Summary of Probability Distributions 483

Appendix C Mathematical Reminders 487

Appendix D Working with Joint Distributions 489

Solutions 497

References 511

Index 515

Robert P. Dobrow, PhD, is Emeritus Professor of Mathematics at Carleton College. He has over 15 years of experience teaching probability and has authored numerous papers in probability theory, Markov chains, and statistics.