One Thousand Exercises in Probability: Third Edition

One Thousand Exercises in Probability: Third Edition

by Geoffrey Grimmett, David Stirzaker

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Overview

This third edition is a revised, updated, and greatly expanded version of previous edition of 2001. The 1300+ exercises contained within are not merely drill problems, but have been chosen to illustrate the concepts, illuminate the subject, and both inform and entertain the reader. A broad range of subjects is covered, including elementary aspects of probability and random variables, sampling, generating functions, Markov chains, convergence, stationary processes, renewals, queues, martingales, diffusions, L vy processes, stability and self-similarity, time changes, and stochastic calculus including option pricing via the Black-Scholes model of mathematical finance.

The text is intended to serve students as a companion for elementary, intermediate, and advanced courses in probability, random processes and operations research. It will also be useful for anyone needing a source for large numbers of problems and questions in these fields. In particular, this book acts as a companion to the authors' volume, Probability and Random Processes, fourth edition (OUP 2020).

Product Details

ISBN-13: 9780198847618
Publisher: Oxford University Press
Publication date: 09/16/2020
Edition description: 3rd ed.
Pages: 592
Sales rank: 692,026
Product dimensions: 9.60(w) x 6.70(h) x 1.10(d)

About the Author

Professor Geoffrey Grimmett, Director of Research and Professor Emeritus of Mathematical Statistics, University of Cambridge, David Stirzaker, Professor Emeritus, Mathematical Institute, University of Oxford

Geoffrey Grimmett is Professor Emeritus of Mathematical Statistics at the University of Cambridge. Cambridge has been his base for pursuing probability theory and the mathematics of disordered systems since 1992. He was Master of Downing College, Cambridge from 2013-2018 and has been appointed Chair of the Heilbronn Institute for Mathematical Research from 2020

He has written numerous research articles in probability theory and statistical mechanics, as well as three research books. With David Stirzaker and Dominic Welsh respectively, he has co-authored two successful textbooks on probability and random processes at the undergraduate and postgraduate levels.

David Stirzaker was educated at Oxford University and Berkeley before being appointed as Fellow and Tutor in Applied Mathematics at St John's College, Oxford. He is now an Emeritus Research Fellow at St John's College, and an Emeritus Professor at the Mathematical Institute, Oxford.

He has written five textbooks on probability and random processes, two of them jointly with Geoffrey Grimmett. Most recently, (2015), he has written The Cambridge Dictionary of Probability and its Applications.

Table of Contents

1 Events and their probabilities

1.1 Introduction

1.2 Events as sets Questions 1, Solutions 181

1.3 Probability Questions 1, Solutions 181

1.4 Conditional probability Questions 2, Solutions 183

1.5 Independence Questions 3, Solutions 185

1.6 Completeness and product spaces

1.7 Worked examples Questions 4, Solutions 186

1.8 Problems Questions 5, Solutions 187

2 Random variables and their distributions

2.1 Random variables Questions 11, Solutions 200

2.2 The law of averages Questions 11, Solutions 201

2.3 Discrete and continuous variables Questions 12, Solutions 201

2.4 Worked examples Questions 12, Solutions 201

2.5 Random vectors Questions 13, Solutions 202

2.6 Monte Carlo simulation

2.7 Problems Questions 13, Solutions 203

3 Discrete random variables

3.1 Probability mass functions Questions 17, Solutions 208

3.2 Independence Questions 17, Solutions 208

3.3 Expectation Questions 18, Solutions 211

3.4 Indicators and matching Questions 20, Solutions 213

3.5 Examples of discrete variables Questions 21, Solutions 218

3.6 Dependence Questions 22, Solutions 218

3.7 Conditional distributions and conditional expectation Questions 23, Solutions 220

3.8 Sums of random variables Questions 24, Solutions 223

3.9 Simple random walk Questions 25, Solutions 225

3.10 Random walk: counting sample paths Questions 26, Solutions 227

3.11 Problems Questions 27, Solutions 228

4 Continuous random variables

4.1 Probability density functions Questions 38, Solutions 251

4.2 Independence Questions 38, Solutions 255

4.3 Expectation Questions 39, Solutions 253

4.4 Examples of continuous variables Questions 41, Solutions 256

4.5 Dependence Questions 42, Solutions 260

4.6 Conditional distributions and conditional expectation Questions 45, Solutions 265

4.7 Functions of random variables Questions 46, Solutions 269

4.8 Sums of random variables Questions 49, Solutions 276

4.9 Multivariate normal distribution Questions 50, Solutions 279

4.10 Distributions arising from the normal distribution Questions 51, Solutions 281

4.11 Sampling from a distribution Questions 51, Solutions 283

4.12 Coupling and Poisson approximation Questions 53, Solutions 286

4.13 Geometrical probability Questions 54, Solutions 287

4.14 Problems Questions 56, Solutions 291

5 Generating functions and their applications

5.1 Generating functions Questions 68, Solutions 318

5.2 Some applications Questions 69, Solutions 321

5.3 Random walk Questions 71, Solutions 324

5.4 Branching processes Questions 72, Solutions 328

5.5 Age-dependent branching processes Questions 73, Solutions 330

5.6 Expectation revisited Questions 74, Solutions 331

5.7 Characteristic functions Questions 74, Solutions 333

5.8 Examples of characteristic functions Questions 76, Solutions 336

5.9 Inversion and continuity theorems Questions 78, Solutions 341

5.10 Two limit theorems Questions 79, Solutions 343

5.11 Large deviations Questions 81, Solutions 348

5.12 Problems Questions 82, Solutions 349

6 Markov chains

6.1 Markov processes Questions 90, Solutions 371

6.2 Classification of states Questions 91, Solutions 375

6.3 Classification of chains Questions 92, Solutions 376

6.4 Stationary distributions and the limit theorem Questions 93, Solutions 380

6.5 Reversibility Questions 96, Solutions 387

6.6 Chains with finitely many states Questions 97, Solutions 389

6.7 Branching processes revisited Questions 99, Solutions 392

6.8 Birth processes and the Poisson process Questions 99, Solutions 393

6.9 Continuous-time Markov chains Questions 101, Solutions 397

6.10 Kolmogorov equations and the limit theorem Questions 102, Solutions 401

6.11 Birth-death processes and imbedding Questions 103, Solutions 403

6.12 Special processes Questions 104, Solutions 406

6.13 Spatial Poisson processes Questions 105, Solutions 407

6.14 Markov chain Monte Carlo Questions 106, Solutions 410

6.15 Problems Questions 107, Solutions 411

7 Convergence of random variables

7.1 Introduction Questions 117, Solutions 433

7.2 Modes of convergence Questions 117, Solutions 433

7.3 Some ancillary results Questions 119, Solutions 437

7.4 Laws of large numbers Questions 120, Solutions 441

7.5 The strong law Questions 121, Solutions 442

7.6 The law of the iterated logarithm Questions 121, Solutions 443

7.7 Martingales Questions 122, Solutions 443

7.8 Martingale convergence theorem Questions 122, Solutions 444

7.9 Prediction and conditional expectation Questions 123, Solutions 445

7.10 Uniform integrability Questions 124, Solutions 448

7.11 Problems Questions 125, Solutions 449

8 Random processes

8.1 Introduction

8.2 Stationary processes Questions 131, Solutions 465

8.3 Renewal processes Questions 131, Solutions 466

8.4 Queues Questions 132, Solutions 467

8.5 The Wiener process Questions 133, Solutions 468

8.6 Levy processes and subordinators Questions 134, Solutions 470

8.7 Self-similarity and stability Questions 134, Solutions 471

8.8 Time changes Questions 134, Solutions 472

8.9 Existence of processes

8.10 Problems Questions 135, Solutions 473

9 Stationary processes

9.1 Introduction Questions 137, Solutions 476

9.2 Linear prediction Questions 138, Solutions 478

9.3 Autocovariances and spectra Questions 138, Solutions 479

9.4 Stochastic integration and the spectral representation Questions 139, Solutions 482

9.5 The ergodic theorem Questions 140, Solutions 482

9.6 Gaussian processes Questions 140, Solutions 483

9.7 Problems Questions 141, Solutions 484

10 Renewals

10.1 The renewal equation Questions 145, Solutions 495

10.2 Limit theorems Questions 146, Solutions 497

10.3 Excess life Questions 146, Solutions 498

10.4 Applications Questions 147, Solutions 500

10.5 Renewal-reward processes 147 502

10.6 Problems Questions 148, Solutions 504

11 Queues

11.1 Single-server queues

11.2 M/M/1 Questions 152, Solutions 510

11.3 M/G/1 Questions 153, Solutions 512

11.4 G/M/1 Questions 153, Solutions 512

11.5 G/G/1 Questions 154, Solutions 513

11.6 Heavy traffic Questions 154, Solutions 514

11.7 Networks of queues Questions 154, Solutions 514

11.8 Problems Questions 155, Solutions 516

12 Martingales

12.1 Introduction Questions 159, Solutions 525

12.2 Martingale differences and Hoeffding's inequality Questions 160, Solutions 527

12.3 Crossings and convergence Questions 160, Solutions 528

12.4 Stopping times Questions 161, Solutions 529

12.5 Optional stopping Questions 162, Solutions 531

12.6 The maximal inequality Questions 163, Solutions 534

12.7 Backward martingales and continuous-time martingales Questions 163, Solutions 534

12.8 Some examples

12.9 Problems Questions 164, Solutions 536

13 Diffusion processes

13.1 Introduction

13.2 Brownian motion Questions 170, Solutions 545

13.3 Diffusion processes Questions 170, Solutions 545

13.4 First passage times Questions 171, Solutions 547

13.5 Barriers Questions 172, Solutions 549

13.6 Excursions and the Brownian bridge Questions 172, Solutions 550

13.7 Stochastic calculus Questions 173, Solutions 551

13.8 The Ito integral Questions 173, Solutions 553

13.9 Ito's formula Questions 174, Solutions 554

13.10 Option pricing Questions 175, Solutions 555

13.11 Passage probabilities and potentials Questions 176, Solutions 556

13.12 Problems Questions 176, Solutions 557

Bibliography 567

Index 569

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