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Introduction toProbability ModelsNinth Edition
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- Page 4 and 5: Introduction toProbability ModelsNi
- Page 6 and 7: ContentsPreface xiii1. Introduction
- Page 8 and 9: Contentsvii4. Markov Chains 1854.1.
- Page 10 and 11: Contentsix8.3. Exponential Models 4
- Page 12 and 13: Contentsxi11.5. Stochastic Processe
- Page 14 and 15: PrefaceThis text is intended as an
- Page 16 and 17: PrefacexvExamples and ExercisesMany
- Page 18 and 19: Prefacexviiimportant simulation tec
- Page 20 and 21: Introduction toProbability Theory11
- Page 22 and 23: 1.2. Sample Space and Events 33 ′
- Page 24 and 25: 1.3. Probabilities Defined on Event
- Page 26 and 27: 1.4. Conditional Probabilities1.4.
- Page 28 and 29: 1.4. Conditional Probabilities 9Sol
- Page 30 and 31: 1.5. Independent Events 11will be q
- Page 32 and 33: 1.6. Bayes’ Formula 13Solution: L
- Page 34 and 35: Exercises 15Example 1.15 You know t
- Page 36 and 37: Exercises 1714. The probability of
- Page 38 and 39: Exercises 19*32. Suppose all n men
- Page 40 and 41: References 21*48. Sixty percent of
- Page 42 and 43: Random Variables22.1. Random Variab
- Page 44 and 45: 2.1. Random Variables 25As a check,
- Page 46 and 47: 2.2. Discrete Random Variables 27(i
- Page 48 and 49: 2.2. Discrete Random Variables 29A
- Page 50 and 51: 2.2. Discrete Random Variables 31wh
- Page 52 and 53:
2.2. Discrete Random Variables 33No
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2.3. Continuous Random Variables 35
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2.3. Continuous Random Variables 37
- Page 58 and 59:
2.4. Expectation of a Random Variab
- Page 60 and 61:
2.4. Expectation of a Random Variab
- Page 62 and 63:
2.4. Expectation of a Random Variab
- Page 64 and 65:
2.4. Expectation of a Random Variab
- Page 66 and 67:
2.5. Jointly Distributed Random Var
- Page 68 and 69:
2.5. Jointly Distributed Random Var
- Page 70 and 71:
2.5. Jointly Distributed Random Var
- Page 72 and 73:
2.5. Jointly Distributed Random Var
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2.5. Jointly Distributed Random Var
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2.5. Jointly Distributed Random Var
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2.5. Jointly Distributed Random Var
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= f(x)=n∑k=i− f(x)2.5. Jointly
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2.5. Jointly Distributed Random Var
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2.6. Moment Generating Functions 65
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2.6. Moment Generating Functions 67
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2.6. Moment Generating Functions 69
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2.6. Moment Generating Functions 71
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2.6. Moment Generating Functions 73
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2.6. Moment Generating Functions 75
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2.7. Limit Theorems 772.7. Limit Th
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2.7. Limit Theorems 79Theorem 2.1 (
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2.7. Limit Theorems 81Table 2.3Area
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2.8. Stochastic Processes 83Taking
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Exercises 85order to visit i + 1 be
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Exercises 8710. Suppose three fair
- Page 108 and 109:
Exercises 89*27. A fair coin is ind
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Exercises 9140. Suppose that two te
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Exercises 93Suppose that E[X]=E[Y ]
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Exercises 95(i) Compute P {X = i}.(
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Conditional Probabilityand Conditio
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3.2. The Discrete Case 99Similarly,
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3.2. The Discrete Case 101Solution:
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3.3. The Continuous Case 103such th
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3.4. Computing Expectations by Cond
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3.4. Computing Expectations by Cond
- Page 128 and 129:
3.4. Computing Expectations by Cond
- Page 130 and 131:
3.4. Computing Expectations by Cond
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3.4. Computing Expectations by Cond
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3.4. Computing Expectations by Cond
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3.4. Computing Expectations by Cond
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3.4. Computing Expectations by Cond
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3.5. Computing Probabilities by Con
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3.5. Computing Probabilities by Con
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3.5. Computing Probabilities by Con
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3.5. Computing Probabilities by Con
- Page 148 and 149:
3.5. Computing Probabilities by Con
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3.5. Computing Probabilities by Con
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3.5. Computing Probabilities by Con
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3.5. Computing Probabilities by Con
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3.6. Some Applications 137The condi
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3.6. Some Applications 139To comput
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3.6. Some Applications 141Now given
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3.6. Some Applications 143Hence,∞
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3.6. Some Applications 145Figure 3.
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Hence, because there are ( nk)subse
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3.6. Some Applications 149(ii) In t
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3.6. Some Applications 151the (n+1)
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3.6. Some Applications 153sis more
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3.6. Some Applications 155to relate
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3.6. Some Applications 157Propositi
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3.7. An Identity for Compound Rando
- Page 180 and 181:
3.7. An Identity for Compound Rando
- Page 182 and 183:
3.7. An Identity for Compound Rando
- Page 184 and 185:
P r (2) ==P r (3) =r(1 − p) [α1
- Page 186 and 187:
Exercises 16716. The random variabl
- Page 188 and 189:
Exercises 169value occurs is(10 9
- Page 190 and 191:
Exercises 171Hint:Is it useful to k
- Page 192 and 193:
Exercises 173will return to its ini
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Exercises 17551. Do Exercise 50 und
- Page 196 and 197:
Exercises 177(b) Find the expected
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Exercises 179Figure 3.7.*73. Suppos
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Exercises 181(c) Prove the followin
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Exercises 183(a) N = min(n: X n−2
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Markov Chains44.1. IntroductionIn t
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4.1. Introduction 187If we let the
- Page 208 and 209:
4.2. Chapman-Kolmogorov Equations 1
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4.2. Chapman-Kolmogorov Equations 1
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4.3. Classification of States 193Su
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4.3. Classification of States 195Ex
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4.3. Classification of States 197Co
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4.3. Classification of States 199wh
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4.3. Classification of States 201α
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4.3. Classification of States 203(i
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4.4. Limiting Probabilities 205We a
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4.4. Limiting Probabilities 207Exam
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4.4. Limiting Probabilities 209Supp
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4.4. Limiting Probabilities 211Henc
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4.4. Limiting Probabilities 213the
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4.4. Limiting Probabilities 215For
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4.5. Some Applications 217orπ 2 =
- Page 238 and 239:
4.5. Some Applications 219and hence
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4.5. Some Applications 221M before
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4.5. Some Applications 223state n a
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4.5. Some Applications 225state i +
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4.5. Some Applications 227implying
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4.5. Some Applications 229given Boo
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4.6. Mean Time Spent in Transient S
- Page 252 and 253:
4.7. Branching Processes 2334.7. Br
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4.7. Branching Processes 235Therefo
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4.8. Time Reversible Markov Chains
- Page 258 and 259:
4.8. Time Reversible Markov Chains
- Page 260 and 261:
4.8. Time Reversible Markov Chains
- Page 262 and 263:
4.8. Time Reversible Markov Chains
- Page 264 and 265:
4.8. Time Reversible Markov Chains
- Page 266 and 267:
4.9. Markov Chain Monte Carlo Metho
- Page 268 and 269:
4.9. Markov Chain Monte Carlo Metho
- Page 270 and 271:
4.9. Markov Chain Monte Carlo Metho
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4.10. Markov Decision Processes 253
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4.10. Markov Decision Processes 255
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4.11. Hidden Markov Chains 257Examp
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4.11. Hidden Markov Chains 259Solut
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4.11. Hidden Markov Chains 261= ∑
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Exercises 263Starting withV 1 (j) =
- Page 284 and 285:
Exercises 2659. Suppose in Exercise
- Page 286 and 287:
Exercises 267Hint: Define an approp
- Page 288 and 289:
Exercises 269ing day with probabili
- Page 290 and 291:
Exercises 271(b) For any state i 1
- Page 292 and 293:
Exercises 27353. Find the average p
- Page 294 and 295:
Exercises 275*62. In Exercise 21,(a
- Page 296 and 297:
Exercises 277Hint:Fix i and show th
- Page 298 and 299:
Exercises 279(c) Let {y ja } be a s
- Page 300 and 301:
The ExponentialDistribution andthe
- Page 302 and 303:
5.2. The Exponential Distribution 2
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5.2. The Exponential Distribution 2
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5.2. The Exponential Distribution 2
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5.2. The Exponential Distribution 2
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5.2. The Exponential Distribution 2
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5.2. The Exponential Distribution 2
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5.2. The Exponential Distribution 2
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5.2. The Exponential Distribution 2
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5.2. The Exponential Distribution 2
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5.2. The Exponential Distribution 3
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5.3. The Poisson Process 303(c) If
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5.3. The Poisson Process 305(ii) Th
- Page 326 and 327:
5.3. The Poisson Process 307Figure
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5.3. The Poisson Process 309That is
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5.3. The Poisson Process 311by cond
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5.3. The Poisson Process 313abiliti
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5.3. The Poisson Process 315Therefo
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5.3. The Poisson Process 317pendent
- Page 338 and 339:
5.3. The Poisson Process 319Example
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5.3. The Poisson Process 321Figure
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5.3. The Poisson Process 323As a re
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5.3. The Poisson Process 325event i
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5.3. The Poisson Process 327Therefo
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5.3. The Poisson Process 329one err
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5.4. Generalizations of the Poisson
- Page 352 and 353:
5.4. Generalizations of the Poisson
- Page 354 and 355:
5.4. Generalizations of the Poisson
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5.4. Generalizations of the Poisson
- Page 358 and 359:
5.4. Generalizations of the Poisson
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5.4. Generalizations of the Poisson
- Page 362 and 363:
5.4. Generalizations of the Poisson
- Page 364 and 365:
5.4. Generalizations of the Poisson
- Page 366 and 367:
Exercises 347suppose that ∑ ni=1
- Page 368 and 369:
Exercises 34921. In a certain syste
- Page 370 and 371:
Exercises 35132. There are three jo
- Page 372 and 373:
Exercises 35344. Cars pass a certai
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Exercises 35554. A viral linear DNA
- Page 376 and 377:
Exercises 357Let X 1 denote the num
- Page 378 and 379:
Exercises 35970. For the infinite s
- Page 380 and 381:
Exercises 361counted events from a
- Page 382 and 383:
Exercises 36389. Some components of
- Page 384 and 385:
Continuous-TimeMarkov Chains66.1. I
- Page 386 and 387:
6.2. Continuous-Time Markov Chains
- Page 388 and 389:
6.3. Birth and Death Processes 369A
- Page 390 and 391:
Putting this back in terms of M(t)
- Page 392 and 393:
6.3. Birth and Death Processes 373(
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6.4. The Transition Probability Fun
- Page 396 and 397:
6.4. The Transition Probability Fun
- Page 398 and 399:
6.4. The Transition Probability Fun
- Page 400 and 401:
6.4. The Transition Probability Fun
- Page 402 and 403:
6.4. The Transition Probability Fun
- Page 404 and 405:
6.5. Limiting Probabilities 385To d
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6.5. Limiting Probabilities 387Solv
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6.5. Limiting Probabilities 389λ n
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6.5. Limiting Probabilities 391pres
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6.6. Time Reversibility 393intuitiv
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6.6. Time Reversibility 395Figure 6
- Page 416 and 417:
6.6. Time Reversibility 397as follo
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6.6. Time Reversibility 399Another
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6.7. Uniformization 4016.7. Uniform
- Page 422 and 423:
6.7. Uniformization 403state 1, it
- Page 424 and 425:
6.8. Computing the Transition Proba
- Page 426 and 427:
Exercises 407raising that matrix to
- Page 428 and 429:
Exercises 409max(X 1 ,...,X j ) can
- Page 430 and 431:
Exercises 41120. There are two mach
- Page 432 and 433:
Exercises 41331. A total of N custo
- Page 434 and 435:
References 41541. Let Y denote an e
- Page 436 and 437:
Renewal Theory andIts Applications7
- Page 438 and 439:
7.2. Distribution of N(t) 419Theref
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7.2. Distribution of N(t) 421Exampl
- Page 442 and 443:
7.3. Limit Theorems and Their Appli
- Page 444 and 445:
7.3. Limit Theorems and Their Appli
- Page 446 and 447:
7.3. Limit Theorems and Their Appli
- Page 448 and 449:
7.3. Limit Theorems and Their Appli
- Page 450 and 451:
7.3. Limit Theorems and Their Appli
- Page 452 and 453:
7.4. Renewal Reward Processes 433Th
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7.4. Renewal Reward Processes 435If
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7.4. Renewal Reward Processes 437re
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7.4. Renewal Reward Processes 439Re
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7.4. Renewal Reward Processes 441wh
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7.5. Regenerative Processes 443Rema
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7.5. Regenerative Processes 445The
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7.5. Regenerative Processes 447smal
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7.5. Regenerative Processes 449Exam
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7.5. Regenerative Processes 451Assu
- Page 472 and 473:
7.6. Semi-Markov Processes 453N∑
- Page 474 and 475:
7.7. The Inspection Paradox 4557.7.
- Page 476 and 477:
7.7. The Inspection Paradox 457is a
- Page 478 and 479:
7.8. Computing the Renewal Function
- Page 480 and 481:
7.9. Applications to Patterns 461Ta
- Page 482 and 483:
7.9. Applications to Patterns 463Th
- Page 484 and 485:
Case 2:In this case,The Overlap Is
- Page 486 and 487:
7.9. Applications to Patterns 467Re
- Page 488 and 489:
7.9. Applications to Patterns 469Su
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7.9. Applications to Patterns 471it
- Page 492 and 493:
7.10. The Insurance Ruin Problem 47
- Page 494 and 495:
7.10. The Insurance Ruin Problem 47
- Page 496 and 497:
7.10. The Insurance Ruin Problem 47
- Page 498 and 499:
Exercises 479Exercises1. Is it true
- Page 500 and 501:
Exercises 48111. A renewal process
- Page 502 and 503:
Exercises 483(c) Suppose that the X
- Page 504 and 505:
Exercises 485(b) Is the reward proc
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Exercises 487(a) Find the long run
- Page 508 and 509:
Exercises 489park your car in Berke
- Page 510 and 511:
Exercises 49151. In 1984 the countr
- Page 512 and 513:
QueueingTheory88.1. IntroductionIn
- Page 514 and 515:
8.2. Preliminaries 4958.2.1. Cost E
- Page 516 and 517:
8.2. Preliminaries 497Two other set
- Page 518 and 519:
8.3. Exponential Models 4998.3. Exp
- Page 520 and 521:
8.3. Exponential Models 501To deter
- Page 522 and 523:
8.3. Exponential Models 503where th
- Page 524 and 525:
8.3. Exponential Models 505amount o
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8.3. Exponential Models 507∞∑P
- Page 528 and 529:
8.3. Exponential Models 509balance
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8.3. Exponential Models 511Now, pot
- Page 532 and 533:
8.3. Exponential Models 513Figure 8
- Page 534 and 535:
8.3. Exponential Models 515she is s
- Page 536 and 537:
8.4. Network of Queues 517Note that
- Page 538 and 539:
8.4. Network of Queues 519To verify
- Page 540 and 541:
8.4. Network of Queues 521The avera
- Page 542 and 543:
8.4. Network of Queues 523Hence, fr
- Page 544 and 545:
8.4. Network of Queues 525where the
- Page 546 and 547:
8.4. Network of Queues 527where a j
- Page 548 and 549:
8.5. The System M/G/1 5298.5.2. App
- Page 550 and 551:
8.6. Variations on the M/G/1 531How
- Page 552 and 553:
8.6. Variations on the M/G/1 533Sub
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8.6. Variations on the M/G/1 535the
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8.6. Variations on the M/G/1 537var
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8.6. Variations on the M/G/1 539fro
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8.6. Variations on the M/G/1 541tom
- Page 562 and 563:
8.7. The Model G/M/1 543These quant
- Page 564 and 565:
8.7. The Model G/M/1 545(We have no
- Page 566 and 567:
8.7. The Model G/M/1 547rate at whi
- Page 568 and 569:
8.8. A Finite Source Model 5498.8.
- Page 570 and 571:
8.8. A Finite Source Model 551Howev
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8.9. Multiserver Queues 5538.9.1. E
- Page 574 and 575:
8.9. Multiserver Queues 555To deriv
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8.9. Multiserver Queues 557distribu
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Exercises 5596. Two customers move
- Page 580 and 581:
Exercises 561(ii) One counter is op
- Page 582 and 583:
Exercises 563In terms of the soluti
- Page 584 and 585:
Exercises 565(b) In terms of the li
- Page 586 and 587:
Exercises 567(a) What is the averag
- Page 588 and 589:
Exercises 56944. Consider the prior
- Page 590 and 591:
Reliability Theory99.1. Introductio
- Page 592 and 593:
9.2. Structure Functions 573Figure
- Page 594 and 595:
9.2. Structure Functions 575Example
- Page 596 and 597:
9.2. Structure Functions 577Figure
- Page 598 and 599:
9.3. Reliability of Systems of Inde
- Page 600 and 601:
9.3. Reliability of Systems of Inde
- Page 602 and 603:
9.4. Bounds on the Reliability Func
- Page 604 and 605:
9.4. Bounds on the Reliability Func
- Page 606 and 607:
9.4. Bounds on the Reliability Func
- Page 608 and 609:
9.4. Bounds on the Reliability Func
- Page 610 and 611:
9.4. Bounds on the Reliability Func
- Page 612 and 613:
9.4. Bounds on the Reliability Func
- Page 614 and 615:
9.5. System Life as a Function of C
- Page 616 and 617:
9.5. System Life as a Function of C
- Page 618 and 619:
9.5. System Life as a Function of C
- Page 620 and 621:
9.5. System Life as a Function of C
- Page 622 and 623:
9.5. System Life as a Function of C
- Page 624 and 625:
9.6. Expected System Lifetime 605an
- Page 626 and 627:
9.6. Expected System Lifetime 607Ma
- Page 628 and 629:
9.6. Expected System Lifetime 609Th
- Page 630 and 631:
9.7. Systems with Repair 611we have
- Page 632 and 633:
9.7. Systems with Repair 613∑ ni=
- Page 634 and 635:
9.7. Systems with Repair 615The pre
- Page 636 and 637:
Exercises 617Since the system fails
- Page 638 and 639:
Exercises 6195. Find the minimal pa
- Page 640 and 641:
Exercises 62113. Let r(p) be the re
- Page 642 and 643:
Exercises 623(a) ¯F t (a) = ¯F(t
- Page 644 and 645:
Brownian Motion andStationary Proce
- Page 646 and 647:
10.1. Brownian Motion 627the molecu
- Page 648 and 649:
10.2. Hitting Times, Maximum Variab
- Page 650 and 651:
10.3. Variations on Brownian Motion
- Page 652 and 653:
10.4. Pricing Stock Options 633Figu
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10.4. Pricing Stock Options 635the
- Page 656 and 657:
10.4. Pricing Stock Options 637Sinc
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10.4. Pricing Stock Options 639eral
- Page 660 and 661:
10.4. Pricing Stock Options 641wher
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10.4. Pricing Stock Options 643for
- Page 664 and 665:
10.5. White Noise 645we see that∫
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10.6. Gaussian Processes 647standar
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10.7. Stationary and Weakly Station
- Page 670 and 671:
10.7. Stationary and Weakly Station
- Page 672 and 673:
10.7. Stationary and Weakly Station
- Page 674 and 675:
10.8. Harmonic Analysis of Weakly S
- Page 676 and 677:
Exercises 657Hence, from Equation (
- Page 678 and 679:
Exercises 659Assume that the contin
- Page 680 and 681:
Exercises 66125. Compute the mean a
- Page 682 and 683:
Simulation1111.1. IntroductionLet X
- Page 684 and 685:
11.1. Introduction 665Table 11.1A R
- Page 686 and 687:
11.1. Introduction 667in estimating
- Page 688 and 689:
11.2. General Techniques for Simula
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11.2. General Techniques for Simula
- Page 692 and 693:
11.2. General Techniques for Simula
- Page 694 and 695:
11.2. General Techniques for Simula
- Page 696 and 697:
11.3. Special Techniques for Simula
- Page 698 and 699:
11.3. Special Techniques for Simula
- Page 700 and 701:
11.3. Special Techniques for Simula
- Page 702 and 703:
11.3. Special Techniques for Simula
- Page 704 and 705:
11.4. Simulating from Discrete Dist
- Page 706 and 707:
11.4. Simulating from Discrete Dist
- Page 708 and 709:
11.4. Simulating from Discrete Dist
- Page 710 and 711:
11.4. Simulating from Discrete Dist
- Page 712 and 713:
11.5. Stochastic Processes 693distr
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11.5. Stochastic Processes 695Now s
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11.5. Stochastic Processes 697where
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11.5. Stochastic Processes 699To st
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11.5. Stochastic Processes 701Propo
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11.6. Variance Reduction Techniques
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11.6. Variance Reduction Techniques
- Page 726 and 727:
11.6. Variance Reduction Techniques
- Page 728 and 729:
11.6. Variance Reduction Techniques
- Page 730 and 731:
11.6. Variance Reduction Techniques
- Page 732 and 733:
11.6. Variance Reduction Techniques
- Page 734 and 735:
11.6. Variance Reduction Techniques
- Page 736 and 737:
11.6. Variance Reduction Techniques
- Page 738 and 739:
11.6. Variance Reduction Techniques
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11.8. Coupling from the Past 721per
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Exercises 723Exercises*1. Suppose i
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Exercises 72510. Explain how we can
- Page 746 and 747:
Exercises 727(e) Use part (d) to sh
- Page 748 and 749:
Exercises 729(a) Show that ∫ X 10
- Page 750 and 751:
References 731If we use simulation
- Page 752 and 753:
AppendixSolutions to StarredExercis
- Page 754 and 755:
30. (a) P {George | exactly 1 hit}=
- Page 756 and 757:
Chapter 2 737Chapter 24. (i) 1, 2,
- Page 758 and 759:
Chapter 2 73972. For the matching p
- Page 760 and 761:
Chapter 3 741p X|Y (3 | 3) = 1 27E[
- Page 762 and 763:
Chapter 3 743sive heads within the
- Page 764 and 765:
Chapter 4 745By symmetry,P nij = 1
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Chapter 5 74762. (a) Since π i = 1
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Chapter 5 749(d) T is the sum of n
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Chapter 6 75184. There is a record
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Chapter 6 753Then this is a birth a
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Chapter 7 75542. (a) The matrix P
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Chapter 7 757ing on the initial mac
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Chapter 8 759Chapter 82. This probl
- Page 780 and 781:
Chapter 8 761(b) The system goes fr
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Chapter 8 76336. The distributions
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Chapter 8 765Hence,= αμ + α E[(d
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Chapter 9 767(b) Suppose λ(t) is i
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Chapter 10 769distribution of X(t)
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Chapter 11 771Chapter 111. (a) Let
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Chapter 11 77323. Let m(t) = ∫ t0
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IndexAbsorbing state of a Markov ch
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Index 777of geometric random variab
- Page 798 and 799:
Index 779tables of, 68-69Monte Carl
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Index 781Poisson, 32-33, 306, 320si
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