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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