Probability and Random ProcessesThe third edition of this successful text gives a rigorous introduction to probability theory and the discussion of the most important random processes in some depth. It includes various topics which are suitable for undergraduate courses, but are not routinely taught. It is suitable to the beginner, and provides a taste and encouragement for more advanced work. There are four main aims: 1) to provide a thorough but straightforward account of basic probability, giving the reader a natural feel for the subject unburdened by oppressive technicalities, 2) to discuss important random processes in depth with many examples. 3) to cover a range of important but less routine topics, 4) to impart to the beginner the flavour of more advanced work. The books begins with basic ideas common to many undergraduate courses in mathematics, statistics and the sciences; in concludes with topics usually found at graduate level. The ordering and numbering of material in this third edition has been mostly preserved from the second. Minor alterations and additions have been added for clearer exposition. Highlights include new sections on sampling and Markov chain Monte Carlo, geometric probability, coupling and Poisson approximation, large deviations, spatial Poisson processes, renewal-reward, queueing networks, stochastic calculus, Itô's formula and option pricing in the Black-Scholes model for financial markets. In addition there are many (nearly 400) new exercises and problems that are entertaining and instructive; their solutions can be found in the companion volume 'One Thousand Exercises in Probability', (OUP 2001). |
Contents
I | 1 |
IV | 4 |
V | 8 |
VI | 13 |
VII | 14 |
VIII | 16 |
IX | 21 |
X | 26 |
LXXIII | 294 |
LXXIV | 303 |
LXXV | 306 |
LXXVI | 316 |
LXXVII | 323 |
LXXVIII | 327 |
LXXIX | 330 |
LXXX | 331 |
XII | 30 |
XIII | 33 |
XIV | 35 |
XV | 38 |
XVI | 41 |
XVII | 43 |
XVIII | 46 |
XX | 48 |
XXI | 50 |
XXII | 56 |
XXIII | 60 |
XXIV | 62 |
XXV | 67 |
XXVI | 70 |
XXVII | 71 |
XXVIII | 75 |
XXIX | 83 |
XXX | 89 |
XXXII | 91 |
XXXIII | 93 |
XXXIV | 95 |
XXXV | 98 |
XXXVI | 104 |
XXXVII | 107 |
XXXVIII | 113 |
XXXIX | 115 |
XL | 119 |
XLI | 122 |
XLII | 127 |
XLIII | 133 |
XLIV | 140 |
XLV | 146 |
XLVII | 154 |
XLVIII | 160 |
XLIX | 169 |
L | 173 |
LI | 176 |
LII | 179 |
LIII | 184 |
LIV | 187 |
LV | 191 |
LVI | 199 |
LVII | 204 |
LVIII | 211 |
LX | 218 |
LXI | 221 |
LXII | 225 |
LXIII | 235 |
LXIV | 238 |
LXV | 241 |
LXVI | 244 |
LXVII | 254 |
LXVIII | 264 |
LXIX | 266 |
LXX | 272 |
LXXI | 279 |
LXXII | 289 |
LXXXI | 336 |
LXXXII | 341 |
LXXXIII | 348 |
LXXXIV | 352 |
LXXXV | 358 |
LXXXVIII | 359 |
LXXXIX | 363 |
XC | 365 |
XCI | 368 |
XCII | 369 |
XCIII | 371 |
XCIV | 371 |
XCVI | 373 |
XCVII | 376 |
XCVIII | 383 |
XCIX | 389 |
C | 401 |
CI | 405 |
CII | 406 |
CIV | 406 |
CV | 407 |
CVI | 409 |
CVII | 417 |
CVIII | 423 |
CIX | 426 |
CXI | 428 |
CXII | 431 |
CXIII | 437 |
CXIV | 441 |
CXV | 448 |
CXVII | 454 |
CXVIII | 457 |
CXXI | 462 |
CXXII | 467 |
CXXIII | 473 |
CXXIV | 477 |
CXXV | 482 |
CXXVI | 485 |
CXXVII | 489 |
CXXVIII | 494 |
CXXIX | 499 |
CXXXI | 500 |
CXXXII | 502 |
CXXXIII | 511 |
CXXXIV | 516 |
CXXXV | 520 |
CXXXVI | 547 |
CXXXVII | 550 |
CXXXVIII | 555 |
CXXXIX | 557 |
CXL | 559 |
CXLII | 562 |
CXLIII | 564 |
CXLIV | 566 |
CXLV | 569 |
CXLVI | 571 |
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Common terms and phrases
arrival branching process called characteristic function conditional expectation continuous continuous-time convergence deduce define Definition denote distribution function distribution with parameter equation ergodic event Example Exercises for Section exists exponentially distributed finite function F Gaussian process given identically distributed random independent identically distributed independent random variables inequality interarrival interval irreducible large numbers Lemma Let X1 Markov chain Markov property martingale mass function non-negative normal distribution notation o-field obtain particle Poisson distribution Poisson process probability generating function probability space Problem Proof queue random walk real numbers renewal process result sample paths satisfies sequence Show standard Wiener process stationary distribution stationary process stochastic strongly stationary subsets Suppose taking values Theorem theory tosses transition matrix vector Wiener process X₁ Y₁ zero means