Introduction to Probability Models

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  • Edition: 10th
  • Format: Hardcover
  • Copyright: 2009-12-03
  • Publisher: Academic Pr
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Ross's classic bestseller, Introduction to Probability Models, has been used extensively by professionals and as the primary text for a first undergraduate course in applied probability. It provides an introduction to elementary probability theory and stochastic processes, and shows how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management science, the physical and social sciences, and operations research. With the addition of several new sections relating to actuaries, this text is highly recommended by the Society of Actuaries. A new section (3.7) on COMPOUND RANDOM VARIABLES, that can be used to establish a recursive formula for computing probability mass functions for a variety of common compounding distributions. A new section (4.11) on HIDDDEN MARKOV CHAINS, including the forward and backward approaches for computing the joint probability mass function of the signals, as well as the Viterbi algorithm for determining the most likely sequence of states. Simplified Approach for Analyzing Nonhomogeneous Poisson processes Additional results on queues relating to the (a) conditional distribution of the number found by an M/M/1 arrival who spends a time t in the system,; (b) inspection paradox for M/M/1 queues (c) M/G/1 queue with server breakdown Many new examples and exercises.

Table of Contents

Prefacep. xi
Introduction to Probability Theoryp. 1
Introductionp. 1
Sample Space and Eventsp. 1
Probabilities Defined on Eventsp. 4
Conditional Probabilitiesp. 7
Independent Eventsp. 10
Bayes' Formulap. 12
Exercisesp. 15
Referencesp. 20
Random Variablesp. 21
Random Variablesp. 21
Discrete Random Variablesp. 25
The Bernoulli Random Variablep. 26
The Binomial Random Variablep. 27
The Geometric Random Variablep. 29
The Poisson Random Variablep. 30
Continuous Random Variablesp. 31
The Uniform Random Variablep. 32
Exponential Random Variablesp. 34
Gamma Random Variablesp. 34
Normal Random Variablesp. 34
Expectation of a Random Variablep. 36
The Discrete Casep. 36
The Continuous Casep. 38
Expectation of a Function of a Random Variablep. 40
Jointly Distributed Random Variablesp. 44
Joint Distribution Functionsp. 44
Independent Random Variablesp. 48
Covariance and Variance of Sums of Random Variablesp. 50
Joint Probability Distribution of Functions of Random Variablesp. 59
Moment Generating Functionsp. 62
The Joint Distribution of the Sample Mean and Sample Variance from a Normal Populationp. 71
The Distribution of the Number of Events that Occurp. 74
Limit Theoremsp. 77
Stochastic Processesp. 84
Exercisesp. 86
Referencesp. 95
Conditional Probability and Conditional Expectationp. 97
Introductionp. 97
The Discrete Casep. 97
The Continuous Casep. 102
Computing Expectations by Conditioningp. 106
Computing Variances by Conditioningp. 117
Computing Probabilities by Conditioningp. 122
Some Applicationsp. 140
A List Modelp. 140
A Random Graphp. 141
Uniform Priors, Polya's Urn Model, and Bose-Einstein Statisticsp. 149
Mean Time for Patternsp. 153
The k-Record Values of Discrete Random Variablesp. 157
Left Skip Free Random Walksp. 160
An Identity for Compound Random Variablesp. 166
Poisson Compounding Distributionp. 169
Binomial Compounding Distributionp. 171
A Compounding Distribution Related to the Negative Binomialp. 172
Exercisesp. 173
Markov Chainsp. 191
Introductionp. 191
Chapman-Kolmogorov Equationsp. 195
Classification of Statesp. 204
Limiting Probabilitiesp. 214
Some Applicationsp. 230
The Gambler's Ruin Problemp. 230
A Model for Algorithmic Efficiencyp. 234
Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problemp. 237
Mean Time Spent in Transient Statesp. 243
Branching Processesp. 245
Time Reversible Markov Chainsp. 249
Markov Chain Monte Carlo Methodsp. 260
Markov Decision Processesp. 265
Hidden Markov Chainsp. 269
Predicting the Statesp. 273
Exercisesp. 275
Referencesp. 290
The Exponential Distribution and the Poisson Processp. 291
Introductionp. 291
The Exponential Distributionp. 292
Definitionp. 292
Properties of the Exponential Distributionp. 294
Further Properties of the Exponential Distributionp. 301
Convolutions of Exponential Random Variablesp. 308
The Poisson Processp. 312
Counting Processesp. 312
Definition of the Poisson Processp. 313
Interarrival and Waiting Time Distributionsp. 316
Further Properties of Poisson Processesp. 319
Conditional Distribution of the Arrival Timesp. 325
Estimating Software Reliabilityp. 336
Generalizations of the Poisson Processp. 339
Nonhomogeneous Poisson Processp. 339
Compound Poisson Processp. 346
Conditional or Mixed Poisson Processesp. 351
Exercisesp. 354
Referencesp. 370
Continuous-Time Markov Chainsp. 371
Introductionp. 371
Continuous-Time Markov Chainsp. 372
Birth and Death Processesp. 374
The Transition Probability Function Pij(t)p. 381
Limiting Probabilitiesp. 390
Time Reversibilityp. 397
Uniformizationp. 406
Computing the Transition Probabilitiesp. 409
Exercisesp. 412
Referencesp. 419
Renewal Theory and Its Applicationsp. 421
Introductionp. 421
Distribution of N(t)p. 423
Limit Theorems and Their Applicationsp. 427
Renewal Reward Processesp. 439
Regenerative Processesp. 447
Alternating Renewal Processesp. 450
Semi-Markov Processesp. 457
The Inspection Paradoxp. 460
Computing the Renewal Functionp. 463
Applications to Patternsp. 466
Patterns of Discrete Random Variablesp. 467
The Expected Time to a Maximal Run of Distinct Valuesp. 474
Increasing Runs of Continuous Random Variablesp. 476
The Insurance Ruin Problemp. 478
Exercisesp. 484
Referencesp. 495
Queueing Theoryp. 497
Introductionp. 497
Preliminariesp. 498
Cost Equationsp. 499
Steady-State Probabilitiesp. 500
Exponential Modelsp. 502
A Single-Server Exponential Queueing Systemp. 502
A Single-Server Exponential Queueing System Having Finite Capacityp. 511
Birth and Death Queueing Modelsp. 517
A Shoe Shine Shopp. 522
A Queueing System with Bulk Servicep. 524
Network of Queuesp. 527
Open Systemsp. 527
Closed Systemsp. 532
The System M/G/1p. 538
Preliminaries: Work and Another Cost Identityp. 538
Application of Work to M/G/1p. 539
Busy Periodsp. 540
Variations on the M/G/1p. 541
The M/G/1 with Random-Sized Batch Arrivalsp. 541
Priority Queuesp. 543
An M/G/1 Optimization Examplep. 546
The M/G/1 Queue with Server Breakdownp. 550
The Model G/M/1p. 553
The G/M/1 Busy and Idle Periodsp. 558
A Finite Source Modelp. 559
Multiserver Queuesp. 562
Erlang's Loss Systemp. 563
The M/M/k Queuep. 564
The G/M/k Queuep. 565
The M/G/k Queuep. 567
Exercisesp. 568
Referencesp. 578
Reliability Theoryp. 579
Introductionp. 579
Structure Functionsp. 580
Minimal Path and Minimal Cut Setsp. 582
Reliability of Systems of Independent Componentsp. 586
Bounds on the Reliability Functionp. 590
Method of Inclusion and Exclusionp. 591
Second Method for Obtaining Bounds on r(p)p. 600
System Life as a Function of Component Livesp. 602
Expected System Lifetimep. 610
An Upper Bound on the Expected Life of a Parallel Systemp. 614
Systems with Repairp. 616
A Series Model with Suspended Animationp. 620
Exercisesp. 623
Referencesp. 629
Brownian Motion and Stationary Processesp. 631
Brownian Motionp. 631
Hitting Times, Maximum Variable, and the Gambler's Ruin Problemp. 635
Variations on Brownian Motionp. 636
Brownian Motion with Driftp. 636
Geometric Brownian Motionp. 636
Pricing Stock Optionsp. 638
An Example in Options Pricingp. 638
The Arbitrage Theoremp. 640
The Black-Scholes Option Pricing Formulap. 644
White Noisep. 649
Gaussian Processesp. 651
Stationary and Weakly Stationary Processesp. 654
Harmonic Analysis of Weakly Stationary Processesp. 659
Exercisesp. 661
Referencesp. 665
Simulationp. 667
Introductionp. 667
General Techniques for Simulating Continuous Random Variablesp. 672
The Inverse Transformation Methodp. 672
The Rejection Methodp. 673
The Hazard Rate Methodp. 677
Special Techniques for Simulating Continuous Random Variablesp. 680
The Normal Distributionp. 680
The Gamma Distributionp. 684
The Chi-Squared Distributionp. 684
The Beta (n, m) Distributionp. 685
The Exponential Distribution-The Von Neumann Algorithmp. 686
Simulating from Discrete Distributionsp. 688
The Alias Methodp. 691
Stochastic Processesp. 696
Simulating a Nonhomogeneous Poisson Processp. 697
Simulating a Two-Dimensional Poisson Processp. 703
Variance Reduction Techniquesp. 706
Use of Antithetic Variablesp. 707
Variance Reduction by Conditioningp. 710
Control Variatesp. 715
Importance Samplingp. 717
Determining the Number of Runsp. 722
Generating from the Stationary Distribution of a Markov Chainp. 723
Coupling from the Pastp. 723
Another Approachp. 725
Exercisesp. 726
Referencesp. 734
Appendix: Solutions to Starred Exercisesp. 735
Indexp. 775
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