Elementary Probability, Second Edition

✍ Scribed by David Stirzaker

• Publisher: Cambridge University Press
• Year: 2003
• Tongue: English
• Leaves: 538
• Edition: 2
• Category: Library

✦ Synopsis

This fully revised and updated new edition of the well established textbook affords a clear introduction to the theory of probability. Topics covered include conditional probability, independence, discrete and continuous random variables, generating functions and limit theorems, and an introduction to Markov chains. The text is accessible to undergraduate students and provides numerous examples and exercises to help develop the important skills necessary for problem solving. First Edition Hb (1994): 0-521-42028-8 First Edition Pb (1994): 0-521-42183-7

✦ Table of Contents

Cover......Page 1
Elementary Probability, 2nd Edition......Page 5
Contents......Page 7
Preface to the Second Edition......Page 13
0.1 Chance......Page 15
0.2 Models......Page 17
0.3 Symmetry......Page 19
0.4 The Long Run......Page 21
0.5 Pay-Offs......Page 22
0.6 Introspection......Page 23
0.7 FAQs......Page 24
0.8 History......Page 28
Notation......Page 29
Sets......Page 30
Venn Diagrams......Page 31
Functions......Page 33
Finite Series......Page 34
Limits......Page 35
Infinite Series......Page 36
1.1 Notation and Experiments......Page 38
1.2 Events......Page 40
1.3 The Addition Rules for Probability......Page 46
1.4 Properties of Probability......Page 48
1.5 Sequences of Events......Page 50
1.6 Remarks......Page 51
Notation......Page 52
Checklist of Terms for Chapter 1......Page 53
1.8 Example: Dice......Page 54
1.9 Example: Urn......Page 55
1.10 Example: Cups and Saucers......Page 56
1.11 Example: Sixes......Page 57
1.12 Example: Family Planning......Page 58
1.13 Example: Craps......Page 59
1.14 Example: Murphy’s Law......Page 60
2.1 Conditional Probability......Page 65
2.2 Independence......Page 71
2.3 Recurrence and Difference Equations......Page 74
2.4 Remarks......Page 76
2.5 Review and Checklist for Chapter 2......Page 78
2.6 Example: Sudden Death......Page 79
2.7 Example: Polya’s Urn......Page 80
2.8 Example: Complacency......Page 81
2.9 Example: Dogfight......Page 82
2.10 Example: Smears......Page 83
2.11 Example: Gambler’s Ruin......Page 84
2.12 Example: Accidents and Insurance......Page 86
Part A: Boys and Girls......Page 87
2.14 Example: Eddington’s Controversy......Page 89
3.1 First Principles......Page 97
3.2 Permutations: Ordered Selection......Page 98
3.3 Combinations: Unordered Selection......Page 100
3.4 Inclusion–Exclusion......Page 101
3.5 Recurrence Relations......Page 102
3.6 Generating Functions......Page 104
3.7 Techniques......Page 107
3.8 Review and Checklist for Chapter 3......Page 109
Checklist of Terms for Chapter 3......Page 110
3.9 Example: Railway Trains......Page 111
3.10 Example: Genoese Lottery......Page 112
3.11 Example: Ringing Birds......Page 113
3.13......Page 115
3.14 Example: Identity......Page 116
3.15 Example: Runs......Page 117
3.16 Example: Fish......Page 119
3.17 Example: Colouring......Page 120
3.18 Example: Matching (Rencontres)......Page 121
4.1 Random Variables......Page 128
4.2 Distributions......Page 129
4.3 Expectation......Page 134
4.4 Conditional Distributions......Page 141
4.5 Sequences of Distributions......Page 144
4.6 Inequalities......Page 145
4.7 Review and Checklist for Chapter 4......Page 148
Checklist of Terms for Chapter 4......Page 150
4.8 Example: Royal Oak Lottery......Page 151
4.9 Example: Misprints......Page 152
4.10 Example: Dog Bites: Poisson Distribution......Page 153
4.11 Example: Guesswork......Page 155
4.12 Example: Gamblers Ruined Again......Page 156
4.13 Example: Postmen......Page 157
4.14 Example: Acme Gadgets......Page 158
4.15 Example: Roulette and the Martingale......Page 159
4.16 Example: Searching......Page 160
4.17 Example: Duelling......Page 161
4.18 Binomial Distribution: The Long Run......Page 163
4.19 Example: Uncertainty and Entropy......Page 164
5.1 Joint Distributions......Page 172
5.2 Independence......Page 176
5.3 Expectation......Page 179
5.4 Sums and Products of Random Variables: Inequalities......Page 186
5.5 Dependence: Conditional Expectation......Page 191
5.6 Simple Random Walk......Page 197
5.7 Martingales......Page 204
5.8 The Law of Averages......Page 210
5.9 Convergence......Page 213
5.10 Review and Checklist for Chapter 5......Page 217
Checklist of Terms for Chapter 5......Page 219
5.11 Example: Golf......Page 220
5.12 Example: Joint Lives......Page 222
5.13 Example: Tournament......Page 223
5.14 Example: Congregations......Page 224
5.15 Example: Propagation......Page 225
5.16 Example: Information and Entropy......Page 226
5.17 Example: Cooperation......Page 228
5.18 Example: Strange But True......Page 229
5.19 Example: Capture–Recapture......Page 230
5.20 Example: Visits of a Random Walk......Page 232
5.21 Example: Ordering......Page 233
5.22 Example: More Martingales......Page 234
5.23 Example: Simple Random Walk Martingales......Page 235
5.24 Example: You Can’t Beat the Odds......Page 236
5.25 Example: Matching Martingales......Page 237
5.26 Example: Three-Handed Gambler’s Ruin......Page 238
6.1 Introduction......Page 246
6.2 Moments and the Probability Generating Function......Page 250
6.3 Sums of Independent Random Variables......Page 253
6.4 Moment Generating Functions......Page 259
6.5 Joint Generating Functions......Page 261
6.6 Sequences......Page 265
6.7 Regeneration......Page 268
6.8 Random Walks......Page 273
6.9 Review and Checklist for Chapter 6......Page 277
Checklist of Terms for Chapter 6......Page 278
Appendix: Calculus......Page 279
Fundamental Theorem of Calculus......Page 280
Functions of More Than One Variable......Page 281
6.10 Example: Gambler’s Ruin and First Passages......Page 282
6.11 Example: “Fair” Pairs of Dice......Page 283
6.12 Example: Branching Process......Page 285
6.13 Example: Geometric Branching......Page 286
6.14 Example: Waring’s Theorem: Occupancy Problems......Page 288
6.15 Example: Bernoulli Patterns and Runs......Page 289
6.16 Example: Waiting for Unusual Light Bulbs......Page 291
6.17 Example: Martingales for Branching......Page 292
6.18 Example: Wald’s Identity......Page 293
6.19 Example: Total Population in Branching......Page 294
7.1 Density and Distribution......Page 301
7.2 Functions of Random Variables......Page 311
7.3 Simulation of Random Variables......Page 315
7.4 Expectation......Page 316
7.5 Moment Generating Functions......Page 320
7.6 Conditional Distributions......Page 324
7.7 Ageing and Survival......Page 326
7.8 Stochastic Ordering......Page 328
7.9 Random Points......Page 329
7.10 Review and Checklist for Chapter 7......Page 332
Checklist of Terms for Chapter 7......Page 334
7.11 Example: Using a Uniform Random Variable......Page 335
7.12 Example: Normal Distribution......Page 337
7.13 Example: Bertrand’s Paradox......Page 338
7.14 Example: Stock Control......Page 340
7.15 Example: Obtaining Your Visa......Page 341
7.16 Example: Pirates......Page 343
7.18 Example: Triangles......Page 344
7.19 Example: Stirling’s Formula......Page 346
8.1 Joint Density and Distribution......Page 351
8.2 Change of Variables......Page 356
8.3 Independence......Page 358
8.4 Sums, Products, and Quotients......Page 362
8.5 Expectation......Page 365
8.6 Conditional Density and Expectation......Page 369
8.7 Transformations: Order Statistics......Page 375
8.8 The Poisson Process: Martingales......Page 378
8.9 Two Limit Theorems......Page 382
8.10 Review and Checklist for Chapter 8......Page 385
Checklist of Terms for Chapter 8......Page 388
8.11 Example: Bivariate Normal Density......Page 389
8.12 Example: Partitions......Page 390
8.13 Example: Buffon’s Needle......Page 391
8.14 Example: Targets......Page 393
8.15 Example: Gamma Densities......Page 394
8.16 Example: Simulation – The Rejection Method......Page 395
8.17 Example: The Inspection Paradox......Page 396
8.18 Example: von Neumann’s Exponential Variable......Page 397
8.19 Example: Maximum from Minima......Page 399
8.20 Example: Binormal and Trinormal......Page 401
8.21 Example: Central Limit Theorem......Page 402
8.22 Example: Poisson Martingales......Page 403
8.24 Example: Characteristic Functions......Page 404
9.1 The Markov Property......Page 410
9.2 Transition Probabilities......Page 414
9.3 First Passage Times......Page 420
9.4 Stationary Distributions......Page 426
9.5 The Long Run......Page 432
9.6 Markov Chains with Continuous Parameter......Page 439
9.7 Forward Equations: Poisson and Birth Processes......Page 442
9.8 Forward Equations: Equilibrium......Page 445
9.9 The Wiener Process and Diffusions......Page 450
of Notation and Formulae......Page 463
Checklist of Terms......Page 464
9.11 Example: Crossing a Cube......Page 465
9.12 Example: Reversible Chains......Page 467
9.13 Example: Diffusion Models......Page 468
9.14 Example: The Renewal Chains......Page 470
9.15 Example: Persistence......Page 471
9.16 Example: First Passages and Bernoulli Patterns......Page 473
9.17 Example: Poisson Processes......Page 475
9.18 Example: Decay......Page 476
9.19 Example: Disasters......Page 477
9.20 Example: The General Birth Process......Page 479
9.21 Example: The Birth–Death Process......Page 480
9.22 Example: Wiener Process with Drift......Page 482
9.23 Example: Markov Chain Martingales......Page 483
9.24 Example: Wiener Process Exiting a Strip......Page 484
9.25Example: Arcsine Law for Zeros......Page 485
9.26 Example: Option Pricing: Black–Scholes Formula......Page 486
Exercises......Page 492
Problems......Page 493
Exercises......Page 494
Problems......Page 496
Exercises......Page 498
Problems......Page 499
Exercises......Page 500
Problems......Page 503
Exercises......Page 505
Problems......Page 507
Exercises......Page 509
Problems......Page 510
Exercises......Page 513
Problems......Page 515
Exercises......Page 516
Problems......Page 519
Exercises......Page 521
Problems......Page 525
History......Page 528
ENVOY......Page 530
Index of Notation......Page 529
Index......Page 531