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One Thousand Exercises in Probability: Third Edition
592Overview
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
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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