Elementary Probability

"Now available in a fully revised and updated new edition, this well-established textbook provides a straightforward introduction to the theory of probability. Topics covered include conditional probability, independence, discrete and continuous random variables, basic combinatorics, generating functions and limit theorems, and an introduction to Markov chains. This edition includes an elementary approach to martingales and the theory of Brownian motion, which supply the cornerstones for many topics in modern financial mathematics such as option and derivative pricing. The text is accessible to undergraduate students, and provides numerous worked examples and exercises to help build the important skills necessary for problem solving."--Jacket Cover......Page 1 Half-title......Page 3 Title......Page 5 Copyright......Page 6 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 Inflnite 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 PROBLEMS......Page 61 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 PROBLEMS......Page 90 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 Example: The Ménages Problem......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 PROBLEMS......Page 122 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 PROBLEMS......Page 165 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 PROBLEMS......Page 240 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 PROBLEMS......Page 295 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 PROBLEMS......Page 348 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 PROBLEMS......Page 405 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 Synopsis 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 9.25 Example: Arcsine Law for Zeros......Page 485 9.26 Example: Option Pricing: Black–Scholes Formula......Page 486 PROBLEMS......Page 487 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 Index of Notation......Page 529 ENVOY......Page 530 Index......Page 531 Now Available In A Fully Revised And Updated Second Edition, This Well Established Textbook Provides A Straightforward Introduction To The Theory Of Probability. The Presentation Is Entertaining Without Any Sacrifice Of Rigour; Important Notions Are Covered With The Clarity That The Subject Demands. Topics Covered Include Conditional Probability, Independence, Discrete And Continuous Random Variables, Basic Combinatorics, Generating Functions And Limit Theorems, And An Introduction To Markov Chains. The Text Is Accessible To Undergraduate Students And Provides Numerous Worked Examples And Exercises To Help Build The Important Skills Necessary For Problem Solving. Introduction -- App. Review Of Elementary Mathematical Prerequisites -- 1. Probability -- 2. Conditional Probability And Independence -- 3. Counting -- 4. Random Variables: Distribution And Expectation -- 5. Random Vectors: Independence And Dependence -- 6. Generating Functions And Their Applications -- 7. Continuous Random Variables -- 8. Jointly Continuous Random Variables -- 9. Markov Chains -- App. Solutions And Hints For Selected Exercises And Problems. David Stirzaker. Includes Bibliographical References (p. 514) And Index. Now available in a fully revised and updated new edition, this well established textbook provides a straightforward introduction to the theory of probability. The presentation is entertaining without any sacrifice of rigour; important notions are covered with the clarity that the subject demands. Topics covered include conditional probability, independence, discrete and continuous random variables, basic combinatorics, generating functions and limit theorems, and an introduction to Markov chains. The text is accessible to undergraduate students and provides numerous worked examples and exercises to help build the important skills necessary for problem solving 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 Now available in a fully revised and updated second edition, this well established undergraduate textbook provides a straightforward introduction to the theory of probability. The presentation is entertaining without any sacrifice of rigour, and important notions are covered with the clarity that the subject demands.