What is included with this book?
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 P_{ij}(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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