Summary

A good probability book at the undergraduate level should develop proper problem-solving skills and mathematical maturity; contain a nice mix of theory and application; and be useful in numerous client disciplines (such as computer science, economics, and engineering). It should be written by someone who has consistently taught the course over numerous years and is tolerant of varying levels of student/reader backgrounds. Probability by Robert Dobrow is such a book. Passionate about problem-solving methods and strategies, Dobrow offers guided assistance for techniques that can be generalized to a wide range of situations (e.g. taking complements, working with indicator variables, conditioning, etc.). The author also introduces and, then, emphasizes simulation (by way of the ever-increasing popularity of freeware R) throughout the text in order to illustrate concepts and highlight computational and theoretical results. Real-life date and examples, over 150 applications, and a multitude of simple and provocative exercises are prevalent. Other key features include discussion of probabilistic topics rather than combinatorial ones; multiple "point-of-view" arguments; and an early introduction to random variables.

Author Biography

ROBERT P. DOBROW, PhD, is Professor of Mathematics at Carleton College. He has taught probability for over fifteen years and has authored numerous papers in probability theory, Markov chains, and statistics.

Preface xiii

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 10

1.6 Counting I 12

1.7 Problem solving strategies: complements, inclusion exclusion 15

1.8 Random variables 19

1.9 A closer look at random variables 22

1.10 A first look at simulation 23

1.11 Chapter summary 27

Exercises 29

2 Conditional Probability 35

2.1 Conditional probability 35

2.2 New information changes the sample space 40

2.3 Finding P(A and B) 42

2.4 Conditioning and the law of total probability 50

2.5 Bayes formula and inverting a conditional probability 57

2.6 Chapter summary 63

Exercises 64

3 Independence and Independent Trials 69

3.1 Independence and dependence 69

3.2 Independent random variables 77

3.3 Bernoulli sequences 79

3.4 Counting II 81

3.5 Binomial distribution 90

3.6 Stirling’s approximation 98

3.7 Poisson distribution 98

3.8 Product spaces* 108

3.9 Chapter summary 110

Exercises 112

4 Random Variables 121

4.1 Expectation 123

4.2 Functions of random variables 125

4.3 Joint distributions 130

4.4 Independent random variables 135

4.5 Linearity of expectation 140

4.6 Variance and standard deviation 145

4.7 Covariance and correlation 154

4.8 Conditional distribution 161

4.9 Properties of covariance and correlation* 167

4.10 Expectation of a function of a random variable* 169

4.11 Chapter summary 171

Exercises 174

5 A Bounty of Discrete Distributions 181

5.1 Geometric distribution 181

5.2 Negative binomial—up from the geometric 189

5.3 Hypergeometric—sampling without replacement 195

5.4 From binomial to multinomial 199

5.5 Benford’s law 206

5.6 Chapter summary 209

Exercises 211

6 Continuous Probability 217

6.1 Probability density function 219

6.2 Cumulative distribution function 223

6.3 Uniform distribution 226

6.4 Expectation and variance 228

6.5 Exponential distribution 230

6.6 Functions of random variables I 235

6.7 Joint distributions 242

6.8 Independence 250

6.8.1 Acceptreject method 253

6.9 Covariance, correlation 256

6.10 Functions of random variables II 258

6.11 Geometric probability 264

6.12 Chapter summary 271

Exercises 275

7 Continuous Distributions 283

7.1 Normal distribution 283

7.2 Gamma distribution 299

7.3 Poisson process 305

7.4 Beta distribution 314

7.5 Pareto distribution, power laws and the 8020 rule 318

7.6 Chapter summary 322

Exercises 325

8 Conditional distribution, expectation, variance 331

8.1 Conditional distributions 331

8.2 Discrete and continuous—mixing it up 338

8.3 Conditional expectation 342

8.4 Computing probabilities by conditioning 352

8.5 Conditional variance 355

8.6 Chapter summary 362

Exercises 364

9 Limits 369

9.1 Weak law of large numbers 371

9.2 Strong law of large numbers 377

9.3 Monte Carlo integration 382

9.4 Central limit theorem 386

9.5 Moment generating functions 395

9.6 Chapter summary 402

Exercises 404

10 Additional Topics 411

10.1 Bivariate normal distribution 411

10.2 Transformations of two random variables 419

10.3 Method of moments 423

10.4 Random walk on graphs 425

10.5 Random walks on weighted graphs and Markov chains 433

10.6 From Markov chain to Markov chain Monte Carlo 440

10.7 Chapter summary 452

Exercises 455

Appendix 460

A Getting started with

R 461

B Probability distributions in

R 471

C Summary of probability distributions 473

D Reminders from algebra and calculus 475

E More problems for practice 477

Problem Solutions 483

Bibliography 500

References 501

Index 504

Index 505

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