Introduction to Probability Models

by Ross

• ISBN13: 9780123756862
• ISBN10: 0123756863
• Edition: 10th
• Format: Hardcover
• Copyright: 2009-12-03
• Publisher: Elsevier Science

Summary

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

Preface	p. xi
Introduction to Probability Theory	p. 1
Introduction	p. 1
Sample Space and Events	p. 1
Probabilities Defined on Events	p. 4
Conditional Probabilities	p. 7
Independent Events	p. 10
Bayes' Formula	p. 12
Exercises	p. 15
References	p. 20
Random Variables	p. 21
Random Variables	p. 21
Discrete Random Variables	p. 25
The Bernoulli Random Variable	p. 26
The Binomial Random Variable	p. 27
The Geometric Random Variable	p. 29
The Poisson Random Variable	p. 30
Continuous Random Variables	p. 31
The Uniform Random Variable	p. 32
Exponential Random Variables	p. 34
Gamma Random Variables	p. 34
Normal Random Variables	p. 34
Expectation of a Random Variable	p. 36
The Discrete Case	p. 36
The Continuous Case	p. 38
Expectation of a Function of a Random Variable	p. 40
Jointly Distributed Random Variables	p. 44
Joint Distribution Functions	p. 44
Independent Random Variables	p. 48
Covariance and Variance of Sums of Random Variables	p. 50
Joint Probability Distribution of Functions of Random Variables	p. 59
Moment Generating Functions	p. 62
The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population	p. 71
The Distribution of the Number of Events that Occur	p. 74
Limit Theorems	p. 77
Stochastic Processes	p. 84
Exercises	p. 86
References	p. 95
Conditional Probability and Conditional Expectation	p. 97
Introduction	p. 97
The Discrete Case	p. 97
The Continuous Case	p. 102
Computing Expectations by Conditioning	p. 106
Computing Variances by Conditioning	p. 117
Computing Probabilities by Conditioning	p. 122
Some Applications	p. 140
A List Model	p. 140
A Random Graph	p. 141
Uniform Priors, Polya's Urn Model, and Bose-Einstein Statistics	p. 149
Mean Time for Patterns	p. 153
The k-Record Values of Discrete Random Variables	p. 157
Left Skip Free Random Walks	p. 160
An Identity for Compound Random Variables	p. 166
Poisson Compounding Distribution	p. 169
Binomial Compounding Distribution	p. 171
A Compounding Distribution Related to the Negative Binomial	p. 172
Exercises	p. 173
Markov Chains	p. 191
Introduction	p. 191
Chapman-Kolmogorov Equations	p. 195
Classification of States	p. 204
Limiting Probabilities	p. 214
Some Applications	p. 230
The Gambler's Ruin Problem	p. 230
A Model for Algorithmic Efficiency	p. 234
Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem	p. 237
Mean Time Spent in Transient States	p. 243
Branching Processes	p. 245
Time Reversible Markov Chains	p. 249
Markov Chain Monte Carlo Methods	p. 260
Markov Decision Processes	p. 265
Hidden Markov Chains	p. 269
Predicting the States	p. 273
Exercises	p. 275
References	p. 290
The Exponential Distribution and the Poisson Process	p. 291
Introduction	p. 291
The Exponential Distribution	p. 292
Definition	p. 292
Properties of the Exponential Distribution	p. 294
Further Properties of the Exponential Distribution	p. 301
Convolutions of Exponential Random Variables	p. 308
The Poisson Process	p. 312
Counting Processes	p. 312
Definition of the Poisson Process	p. 313
Interarrival and Waiting Time Distributions	p. 316
Further Properties of Poisson Processes	p. 319
Conditional Distribution of the Arrival Times	p. 325
Estimating Software Reliability	p. 336
Generalizations of the Poisson Process	p. 339
Nonhomogeneous Poisson Process	p. 339
Compound Poisson Process	p. 346
Conditional or Mixed Poisson Processes	p. 351
Exercises	p. 354
References	p. 370
Continuous-Time Markov Chains	p. 371
Introduction	p. 371
Continuous-Time Markov Chains	p. 372
Birth and Death Processes	p. 374
The Transition Probability Function Pij(t)	p. 381
Limiting Probabilities	p. 390
Time Reversibility	p. 397
Uniformization	p. 406
Computing the Transition Probabilities	p. 409
Exercises	p. 412
References	p. 419
Renewal Theory and Its Applications	p. 421
Introduction	p. 421
Distribution of N(t)	p. 423
Limit Theorems and Their Applications	p. 427
Renewal Reward Processes	p. 439
Regenerative Processes	p. 447
Alternating Renewal Processes	p. 450
Semi-Markov Processes	p. 457
The Inspection Paradox	p. 460
Computing the Renewal Function	p. 463
Applications to Patterns	p. 466
Patterns of Discrete Random Variables	p. 467
The Expected Time to a Maximal Run of Distinct Values	p. 474
Increasing Runs of Continuous Random Variables	p. 476
The Insurance Ruin Problem	p. 478
Exercises	p. 484
References	p. 495
Queueing Theory	p. 497
Introduction	p. 497
Preliminaries	p. 498
Cost Equations	p. 499
Steady-State Probabilities	p. 500
Exponential Models	p. 502
A Single-Server Exponential Queueing System	p. 502
A Single-Server Exponential Queueing System Having Finite Capacity	p. 511
Birth and Death Queueing Models	p. 517
A Shoe Shine Shop	p. 522
A Queueing System with Bulk Service	p. 524
Network of Queues	p. 527
Open Systems	p. 527
Closed Systems	p. 532
The System M/G/1	p. 538
Preliminaries: Work and Another Cost Identity	p. 538
Application of Work to M/G/1	p. 539
Busy Periods	p. 540
Variations on the M/G/1	p. 541
The M/G/1 with Random-Sized Batch Arrivals	p. 541
Priority Queues	p. 543
An M/G/1 Optimization Example	p. 546
The M/G/1 Queue with Server Breakdown	p. 550
The Model G/M/1	p. 553
The G/M/1 Busy and Idle Periods	p. 558
A Finite Source Model	p. 559
Multiserver Queues	p. 562
Erlang's Loss System	p. 563
The M/M/k Queue	p. 564
The G/M/k Queue	p. 565
The M/G/k Queue	p. 567
Exercises	p. 568
References	p. 578
Reliability Theory	p. 579
Introduction	p. 579
Structure Functions	p. 580
Minimal Path and Minimal Cut Sets	p. 582
Reliability of Systems of Independent Components	p. 586
Bounds on the Reliability Function	p. 590
Method of Inclusion and Exclusion	p. 591
Second Method for Obtaining Bounds on r(p)	p. 600
System Life as a Function of Component Lives	p. 602
Expected System Lifetime	p. 610
An Upper Bound on the Expected Life of a Parallel System	p. 614
Systems with Repair	p. 616
A Series Model with Suspended Animation	p. 620
Exercises	p. 623
References	p. 629
Brownian Motion and Stationary Processes	p. 631
Brownian Motion	p. 631
Hitting Times, Maximum Variable, and the Gambler's Ruin Problem	p. 635
Variations on Brownian Motion	p. 636
Brownian Motion with Drift	p. 636
Geometric Brownian Motion	p. 636
Pricing Stock Options	p. 638
An Example in Options Pricing	p. 638
The Arbitrage Theorem	p. 640
The Black-Scholes Option Pricing Formula	p. 644
White Noise	p. 649
Gaussian Processes	p. 651
Stationary and Weakly Stationary Processes	p. 654
Harmonic Analysis of Weakly Stationary Processes	p. 659
Exercises	p. 661
References	p. 665
Simulation	p. 667
Introduction	p. 667
General Techniques for Simulating Continuous Random Variables	p. 672
The Inverse Transformation Method	p. 672
The Rejection Method	p. 673
The Hazard Rate Method	p. 677
Special Techniques for Simulating Continuous Random Variables	p. 680
The Normal Distribution	p. 680
The Gamma Distribution	p. 684
The Chi-Squared Distribution	p. 684
The Beta (n, m) Distribution	p. 685
The Exponential Distribution-The Von Neumann Algorithm	p. 686
Simulating from Discrete Distributions	p. 688
The Alias Method	p. 691
Stochastic Processes	p. 696
Simulating a Nonhomogeneous Poisson Process	p. 697
Simulating a Two-Dimensional Poisson Process	p. 703
Variance Reduction Techniques	p. 706
Use of Antithetic Variables	p. 707
Variance Reduction by Conditioning	p. 710
Control Variates	p. 715
Importance Sampling	p. 717
Determining the Number of Runs	p. 722
Generating from the Stationary Distribution of a Markov Chain	p. 723
Coupling from the Past	p. 723
Another Approach	p. 725
Exercises	p. 726
References	p. 734
Appendix: Solutions to Starred Exercises	p. 735
Index	p. 775
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