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# Introduction to Probability Models

**by**Ross, Sheldon M.

10th

### 9780123756862

0123756863

Hardcover

12/3/2009

Academic Pr

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## 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 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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