Introduction to Probability Models

by Sheldon M. Ross

Overview

This best-selling reference is well-suited to those seeking to apply probability theory to phenomena in such fields as engineering, actuarial and management sciences, the physical and social sciences, and operations research.

Realistic models of real-world phenomena must take into account the possibility of randomness. More often than not, quantities are not predictable, but exhibit variations that should be taken into account by the model. This is usually accomplished by allowing the model to be probabilistic in nature. Such a model is referred as a probability model.

Introduction to Probability Models is a fascinating introduction to applications from diverse disciplines and an excellent introduction to a wide variety of applied probability topics.

* Best-selling book by a well-known author, with over 20,000 in sales for 7th edition

* Includes new examples and exercises in actuarial sciences

* Contains compulsory material for Exam 3 of the Society of Actuaries

Author Biography: Sheldon M. Ross is a professor in the Department of Industrial Engineering and Operations Research at the University of California, Berkeley. He received his Ph.D. in statistics at Stanford University in 1968 and has been at Berkeley ever since. He has published many technical articles and textbooks in the areas of statistics and applied probability. Among his texts are A First Course in Probability, Fourth Edition published by MacMillan, Introduction to Probability Models, Fifth Edition published by Academic Press, Stochastic Processes, Second Edition published by Wiley, and a new text, Introductory Statistics published by McGraw Hill.

Professor Ross is the founding and continuing editor of the journal Probability in the Engineering and Informational Sciences published by Cambridge University Press. He is a Fellow of the Institute of Mathematical Statistics, and a recipient of the Humboldt US Senior Scientist Award.

Product Details

ISBN-13:	9780128143476
Publisher:	Elsevier Science
Publication date:	03/09/2019
Sold by:	Barnes & Noble
Format:	eBook
Pages:	842
Sales rank:	900,914
File size:	25 MB
Note:	This product may take a few minutes to download.

About the Author

Dr. Sheldon M. Ross is a professor in the Department of Industrial and Systems Engineering at the University of Southern California. He received his PhD in statistics at Stanford University in 1968. He has published many technical articles and textbooks in the areas of statistics and applied probability. Among his texts are A First Course in Probability, Introduction to Probability Models, Stochastic Processes, and Introductory Statistics. Professor Ross is the founding and continuing editor of the journal Probability in the Engineering and Informational Sciences. He is a Fellow of the Institute of Mathematical Statistics, a Fellow of INFORMS, and a recipient of the Humboldt US Senior Scientist Award.

Table of Contents

	Preface to the Fifth Edition	xi
	Preface to the Sixth Edition	xiii
	Preface to the Seventh Edition	xv
1.	Introduction to Probability Theory	1
1.1.	Introduction	1
1.2.	Sample Space and Events	1
1.3.	Probabilities Defined on Events	4
1.4.	Conditional Probabilities	6
1.5.	Independent Events	10
1.6.	Bayes' Formula	12
	Exercises	15
	References	21
2.	Random Variables	23
2.1.	Random Variables	23
2.2.	Discrete Random Variables	27
2.2.1.	The Bernoulli Random Variable	27
2.2.2.	The Binomial Random Variable	28
2.2.3.	The Geometric Random Variable	31
2.2.4.	The Poisson Random Variable	31
2.3.	Continuous Random Variables	33
2.3.1.	The Uniform Random Variable	34
2.3.2.	Exponential Random Variables	35
2.3.3.	Gamma Random Variables	35
2.3.4.	Normal Random Variables	36
2.4.	Expectation of a Random Variable	37
2.4.1.	The Discrete Case	37
2.4.2.	The Continuous Case	40
2.4.3.	Expectation of a Function of a Random Variable	42
2.5.	Jointly Distributed Random Variables	46
2.5.1.	Joint Distribution Functions	46
2.5.2.	Independent Random Variables	50
2.5.3.	Covariance and Variance of Sums of Random Variables	51
2.5.4.	Joint Probability Distribution of Functions of Random Variables	59
2.6.	Moment Generating Functions	62
2.6.1.	The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population	70
2.7.	Limit Theorems	73
2.8.	Stochastic Processes	79
	Exercises	82
	References	92
3.	Conditional Probability and Conditional Expectation	93
3.1.	Introduction	93
3.2.	The Discrete Case	93
3.3.	The Continuous Case	98
3.4.	Computing Expectations by Conditioning	101
3.5.	Computing Probabilities by Conditioning	114
3.6.	Some Applications	128
3.6.1.	A List Model	128
3.6.2.	A Random Graph	129
3.6.3.	Uniform Priors, Polya's Urn Model, and Bose-Einstein Statistics	137
3.6.4.	The k-Record Values of Discrete Random Variables	141
	Exercises	145
4.	Markov Chains	163
4.1.	Introduction	163
4.2.	Chapman-Kolmogorov Equations	166
4.3.	Classification of States	168
4.4.	Limiting Probabilities	178
4.5.	Some Applications	188
4.5.1.	The Gambler's Ruin Problem	188
4.5.2.	A Model for Algorithmic Efficiency	192
4.5.3.	Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem	194
4.6.	Mean Time Spent in Transient States	200
4.7.	Branching Processes	202
4.8.	Time Reversible Markov Chains	205
4.9.	Markov Chain Monte Carlo Methods	216
4.10.	Markov Decision Processes	222
	Exercises	226
	References	240
5.	The Exponential Distribution and the Poisson Process	241
5.1.	Introduction	241
5.2.	The Exponential Distribution	242
5.2.1.	Definition	242
5.2.2.	Properties of the Exponential Distribution	243
5.2.3.	Further Properties of the Exponential Distribution	248
5.2.4.	Convolutions of Exponential Random Variables	253
5.3.	The Poisson Process	256
5.3.1.	Counting Processes	256
5.3.2.	Definition of the Poisson Process	258
5.3.3.	Interarrival and Waiting Time Distributions	261
5.3.4.	Further Properties of Poisson Processes	264
5.3.5.	Conditional Distribution of the Arrival Times	270
5.3.6.	Estimating Software Reliability	281
5.4.	Generalizations of the Poisson Process	284
5.4.1.	Nonhomogeneous Poisson Process	284
5.4.2.	Compound Poisson Process	289
	Exercises	295
	References	311
6.	Continuous-Time Markov Chains	313
6.1.	Introduction	313
6.2.	Continuous-Time Markov Chains	314
6.3.	Birth and Death Processes	316
6.4.	The Transition Probability Function P[subscript ij](t)	323
6.5.	Limiting Probabilities	331
6.6.	Time Reversibility	338
6.7.	Uniformization	346
6.8.	Computing the Transition Probabilities	349
	Exercises	352
	References	361
7.	Renewal Theory and Its Applications	363
7.1.	Introduction	363
7.2.	Distribution of N(t)	365
7.3.	Limit Theorems and Their Applications	368
7.4.	Renewal Reward Processes	377
7.5.	Regenerative Processes	386
7.5.1.	Alternating Renewal Processes	389
7.6.	Semi-Markov Processes	395
7.7.	The Inspection Paradox	398
7.8.	Computing the Renewal Function	400
7.9.	Applications to Patterns	403
7.9.1.	Patterns of Discrete Random Variables	404
7.9.2.	The Expected Time to a Maximal Run of Distinct Values	410
7.9.3.	Increasing Runs of Continuous Random Variables	412
	Exercises	413
	References	425
8.	Queueing Theory	427
8.1.	Introduction	427
8.2.	Preliminaries	428
8.2.1.	Cost Equations	429
8.2.2.	Steady-State Probabilities	430
8.3.	Exponential Models	432
8.3.1.	A Single-Server Exponential Queueing System	432
8.3.2.	A Single-Server Exponential Queueing System Having Finite Capacity	438
8.3.3.	A Shoeshine Shop	442
8.3.4.	A Queueing System with Bulk Service	444
8.4.	Network of Queues	447
8.4.1.	Open Systems	447
8.4.2.	Closed Systems	452
8.5.	The System M/G/1	458
8.5.1.	Preliminaries: Work and Another Cost Identity	458
8.5.2.	Application of Work to M/G/1	459
8.5.3.	Busy Periods	460
8.6.	Variations on the M/G/1	461
8.6.1.	The M/G/1 with Random-Sized Batch Arrivals	461
8.6.2.	Priority Queues	463
8.6.3.	An M/G/1 Optimization Example	466
8.7.	The Model G/M/1	470
8.7.1.	The G/M/1 Busy and Idle Periods	475
8.8.	A Finite Source Model	475
8.9.	Multiserver Queues	479
8.9.1.	Erlang's Loss System	479
8.9.2.	The M/M/k Queue	481
8.9.3.	The G/M/k Queue	481
8.9.4.	The M/G/k Queue	483
	Exercises	484
	References	496
9.	Reliability Theory	499
9.1.	Introduction	499
9.2.	Structure Functions	500
9.2.1.	Minimal Path and Minimal Cut Sets	502
9.3.	Reliability of Systems of Independent Components	506
9.4.	Bounds on the Reliability Function	510
9.4.1.	Method of Inclusion and Exclusion	511
9.4.2.	Second Method for Obtaining Bounds on r(p)	519
9.5.	System Life as a Function of Component Lives	521
9.6.	Expected System Lifetime	529
9.6.1.	An Upper Bound on the Expected Life of a Parallel System	533
9.7.	Systems with Repair	535
9.7.1.	A Series Model with Suspended Animation	539
	Exercises	542
	References	548
10.	Brownian Motion and Stationary Processes	549
10.1.	Brownian Motion	549
10.2.	Hitting Times, Maximum Variable, and the Gambler's Ruin Problem	553
10.3.	Variations on Brownian Motion	554
10.3.1.	Brownian Motion with Drift	554
10.3.2.	Geometric Brownian Motion	555
10.4.	Pricing Stock Options	556
10.4.1.	An Example in Options Pricing	556
10.4.2.	The Arbitrage Theorem	558
10.4.3.	The Black-Scholes Option Pricing Formula	561
10.5.	White Noise	567
10.6.	Gaussian Processes	569
10.7.	Stationary and Weakly Stationary Processes	572
10.8.	Harmonic Analysis of Weakly Stationary Processes	577
	Exercises	579
	References	584
11.	Simulation	585
11.1.	Introduction	585
11.2.	General Techniques for Simulating Continuous Random Variables	590
11.2.1.	The Inverse Transformation Method	590
11.2.2.	The Rejection Method	591
11.2.3.	The Hazard Rate Method	595
11.3.	Special Techniques for Simulating Continuous Random Variables	598
11.3.1.	The Normal Distribution	598
11.3.2.	The Gamma Distribution	602
11.3.3.	The Chi-Squared Distribution	602
11.3.4.	The Beta (n, m) Distribution	603
11.3.5.	The Exponential Distribution--The Von Neumann Algorithm	604
11.4.	Simulating from Discrete Distributions	606
11.4.1.	The Alias Method	610
11.5.	Stochastic Processes	613
11.5.1.	Simulating a Nonhomogeneous Poisson Process	615
11.5.2.	Simulating a Two-Dimensional Poisson Process	621
11.6.	Variance Reduction Techniques	624
11.6.1.	Use of Antithetic Variables	625
11.6.2.	Variance Reduction by Conditioning	629
11.6.3.	Control Variates	633
11.6.4.	Importance Sampling	634
11.7.	Determining the Number of Runs	639
	Exercises	640
	References	648
Appendix	Solutions to Starred Exercises	649
	Index	687

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Ross's bestseller—now updated with new material for actuarial exams and current applications in engineering, science, business, and economics

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Praise from Reviewers:“I think Ross has done an admirable job of covering the breadth of applied probability. Ross writes fantastic problems which really force the students to think divergently...The examples, like the exercises are great.”- Matt Carlton, California Polytechnic Institute“This is a fascinating introduction to applications from a variety of disciplines. Any curious student will love this book."- Jean LeMaire, University of Pennsylvania“This book may be a model in the organization of the education process. I would definitely rate this text to be the best probability models book at its level of difficulty...far more sophisticated and deliberate than its competitors.”- Kris Ostaszewski, University of Illinois

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