Introduction to Probability Models

by Ross

• ISBN13: 9780125980623
• ISBN10: 0125980620
• eBook ISBN(s): 9780080467825, 9780125980623
• Edition: 9th
• Format: Hardcover
• Copyright: 2006-11-21
• Publisher: Elsevier Science
• More Book Details
• Purchase Benefits

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	xiii
1. Introduction to Probability Theory	1	(22)
1.1. Introduction	1	(1)
1.2. Sample Space and Events	1	(3)
1.3. Probabilities Defined on Events	4	(3)
1.4. Conditional Probabilities	7	(3)
1.5. Independent Events	10	(2)
1.6. Bayes' Formula	12	(3)
Exercises	15	(6)
References	21	(2)
2. Random Variables	23	(74)
2.1. Random Variables	23	(4)
2.2. Discrete Random Variables	27	(7)
2.2.1. The Bernoulli Random Variable	28	(1)
2.2.2. The Binomial Random Variable	29	(2)
2.2.3. The Geometric Random Variable	31	(1)
2.2.4. The Poisson Random Variable	32	(2)
2.3. Continuous Random Variables	34	(4)
2.3.1. The Uniform Random Variable	35	(1)
2.3.2. Exponential Random Variables	36	(1)
2.3.3. Gamma Random Variables	37	(1)
2.3.4. Normal Random Variables	37	(1)
2.4. Expectation of a Random Variable	38	(9)
2.4.1. The Discrete Case	38	(3)
2.4.2. The Continuous Case	41	(2)
2.4.3. Expectation of a Function of a Random Variable	43	(4)
2.5. Jointly Distributed Random Variables	47	(17)
2.5.1. Joint Distribution Functions	47	(4)
2.5.2. Independent Random Variables	51	(2)
2.5.3. Covariance and Variance of Sums of Random Variables	53	(8)
2.5.4. Joint Probability Distribution of Functions of Random Variables	61	(3)
2.6. Moment Generating Functions	64	(13)
2.6.1. The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population	74	(3)
2.7. Limit Theorems	77	(6)
2.8. Stochastic Processes	83	(2)
Exercises	85	(11)
References	96	(1)
3. Conditional Probability and Conditional Expectation	97	(88)
3.1. Introduction	97	(1)
3.2. The Discrete Case	97	(5)
3.3. The Continuous Case	102	(3)
3.4. Computing Expectations by Conditioning	105	(15)
3.4.1. Computing Variances by Conditioning	117	(3)
3.5. Computing Probabilities by Conditioning	120	(17)
3.6. Some Applications	137	(21)
3.6.1. A List Model	137	(2)
3.6.2. A Random Graph	139	(8)
3.6.3. Uniform Priors, Polya's Urn Model, and Bose–Einstein Statistics	147	(4)
3.6.4. Mean Time for Patterns	151	(4)
3.6.5. The k-Record Values of Discrete Random Variables	155	(3)
3.7. An Identity for Compound Random Variables	158	(7)
3.7.1. Poisson Compounding Distribution	161	(2)
3.7.2. Binomial Compounding Distribution	163	(1)
3.7.3. A Compounding Distribution Related to the Negative Binomial	164	(1)
Exercises	165	(20)
4. Markov Chains	185	(96)
4.1. Introduction	185	(4)
4.2. Chapman—Kolmogorov Equations	189	(4)
4.3. Classification of States	193	(11)
4.4. Limiting Probabilities	204	(13)
4.5. Some Applications	217	(13)
4.5.1. The Gambler's Ruin Problem	217	(4)
4.5.2. A Model for Algorithmic Efficiency	221	(3)
4.5.3. Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem	224	(6)
4.6. Mean Time Spent in Transient States	230	(3)
4.7. Branching Processes	233	(3)
4.8. Time Reversible Markov Chains	236	(11)
4.9. Markov Chain Monte Carlo Methods	247	(5)
4.10. Markov Decision Processes	252	(4)
4.11. Hidden Markov Chains	256	(7)
4.11.1. Predicting the States	261	(2)
Exercises	263	(17)
References	280	(1)
5. The Exponential Distribution and the Poisson Process	281	(84)
5.1. Introduction	281	(1)
5.2. The Exponential Distribution	282	(20)
5.2.1. Definition	282	(2)
5.2.2. Properties of the Exponential Distribution	284	(7)
5.2.3. Further Properties of the Exponential Distribution	291	(7)
5.2.4. Convolutions of Exponential Random Variables	298	(4)
5.3. The Poisson Process	302	(28)
5.3.1. Counting Processes	302	(2)
5.3.2. Definition of the Poisson Process	304	(3)
5.3.3. Interarrival and Waiting Time Distributions	307	(3)
5.3.4. Further Properties of Poisson Processes	310	(6)
5.3.5. Conditional Distribution of the Arrival Times	316	(12)
5.3.6. Estimating Software Reliability	328	(2)
5.4. Generalizations of the Poisson Process	330	(16)
5.4.1. Nonhomogeneous Poisson Process	330	(7)
5.4.2. Compound Poisson Process	337	(6)
5.4.3. Conditional or Mixed Poisson Processes	343	(3)
Exercises	346	(18)
References	364	(1)
6. Continuous-Time Markov Chains	365	(52)
6.1. Introduction	365	(1)
6.2. Continuous-Time Markov Chains	366	(2)
6.3. Birth and Death Processes	368	(7)
6.4. The Transition Probability Function PPS (t)	375	(9)
6.5. Limiting Probabilities	384	(8)
6.6. Time Reversibility	392	(9)
6.7. Uniformization	401	(3)
6.8. Computing the Transition Probabilities	404	(3)
Exercises	407	(8)
References	415	(2)
7. Renewal Theory and Its Applications	417	(76)
7.1. Introduction	417	(2)
7.2. Distribution of N (t)	419	(4)
7.3. Limit Theorems and Their Applications	423	(10)
7.4. Renewal Reward Processes	433	(9)
7.5. Regenerative Processes	442	(10)
7.5.1. Alternating Renewal Processes	445	(7)
7.6. Semi-Markov Processes	452	(3)
7.7. The Inspection Paradox	455	(3)
7.8. Computing the Renewal Function	458	(3)
7.9. Applications to Patterns	461	(12)
7.9.1. Patterns of Discrete Random Variables	462	(7)
7.9.2. The Expected Time to a Maximal Run of Distinct Values	469	(2)
7.9.3. Increasing Runs of Continuous Random Variables	471	(2)
7.10. The Insurance Ruin Problem	473	(6)
Exercises	479	(13)
References	492	(1)
8. Queueing Theory	493	(78)
8.1. Introduction	493	(1)
8.2. Preliminaries	494	(5)
8.2.1. Cost Equations	495	(1)
8.2.2. Steady-State Probabilities	496	(3)
8.3. Exponential Models	499	(18)
8.3.1. A Single-Server Exponential Queueing System	499	(9)
8.3.2. A Single-Server Exponential Queueing System Having Finite Capacity	508	(3)
8.3.3. A Shoeshine Shop	511	(3)
8.3.4. A Queueing System with Bulk Service	514	(3)
8.4. Network of Queues	517	(11)
8.4.1. Open Systems	517	(5)
8.4.2. Closed Systems	522	(6)
8.5. The System MIG/1	528	(3)
8.5.1. Preliminaries: Work and Another Cost Identity	528	(1)
8.5.2. Application of Work to M/G/1	529	(1)
8.5.3. Busy Periods	530	(1)
8.6. Variations on the M/G/1	531	(12)
8.6.1. The M/G/1 with Random-Sized Batch Arrivals	531	(2)
8.6.2. Priority Queues	533	(3)
8.6.3. An M/G/1 Optimization Example	536	(4)
8.6.4. The M/G/1 Queue with Server Breakdown	540	(3)
8.7. The Model GIMI1	543	(6)
8.7.1. The G/M/1 Busy and Idle Periods	548	(1)
8.8. A Finite Source Model	549	(3)
8.9. Multiserver Queues	552	(6)
8.9.1. Erlang's Loss System	553	(1)
8.9.2. The M/M/k Queue	554	(1)
8.9.3. The G/M/k Queue	554	(2)
8.9.4. The M/G/k Queue	556	(2)
Exercises	558	(12)
References	570	(1)
9. Reliability Theory	571	(54)
9.1. Introduction	571	(1)
9.2. Structure Functions	571	(7)
9.2.1. Minimal Path and Minimal Cut Sets	574	(4)
9.3 Reliability of Systems of Independent Components	578	(5)
9.4. Bounds on the Reliability Function	583	(12)
9.4.1. Method of Inclusion and Exclusion	584	(9)
9.4.2. Second Method for Obtaining Bounds on r (p)	593	(2)
9.5. System Life as a Function of Component Lives	595	(9)
9.6. Expected System Lifetime	604	(6)
9.6.1. An Upper Bound on the Expected Life of a Parallel System	608	(2)
9.7. Systems with Repair	610	(7)
9.7.1. A Series Model with Suspended Animation	615	(2)
Exercises	617	(7)
References	624	(1)
10. Brownian Motion and Stationary Processes	625	(38)
10.1. Brownian Motion	625	(4)
10.2. Hitting Times, Maximum Variable, and the Gambler's Ruin Problem	629	(2)
10.3. Variations on Brownian Motion	631	(1)
10.3.1. Brownian Motion with Drift	631	(1)
10.3.2. Geometric Brownian Motion	631	(1)
10.4. Pricing Stock Options	632	(12)
10.4.1. An Example in Options Pricing	632	(3)
10.4.2. The Arbitrage Theorem	635	(3)
10.4.3. The Black-Scholes Option Pricing Formula	638	(6)
10.5. White Noise	644	(2)
10.6. Gaussian Processes	646	(3)
10.7. Stationary and Weakly Stationary Processes	649	(5)
10.8. Harmonic Analysis of Weakly Stationary Processes	654	(3)
Exercises	657	(5)
References	662	(1)
11. Simulation	663	(70)
11.1. Introduction	663	(5)
11.2. General Techniques for Simulating Continuous Random Variables	668	(9)
11.2.1. The Inverse Transformation Method	668	(1)
11.2.2. The Rejection Method	669	(4)
11.2.3. The Hazard Rate Method	673	(4)
11.3. Special Techniques for Simulating Continuous Random Variables	677	(8)
11.3.1. The Normal Distribution	677	(3)
11.3.2. The Gamma Distribution	680	(1)
11.3.3. The Chi-Squared Distribution	681	(1)
11.3.4. The Beta (n, m) Distribution	681	(1)
11.3.5. The Exponential Distribution—The Von Neumann Algorithm	682	(3)
11.4. Simulating from Discrete Distributions	685	(7)
11.4.1. The Alias Method	688	(4)
11.5. Stochastic Processes	692	(11)
11.5.1. Simulating a Nonhomogeneous Poisson Process	693	(7)
11.5.2. Simulating a Two-Dimensional Poisson Process	700	(3)
11.6. Variance Reduction Techniques	703	(17)
11.6.1. Use of Antithetic Variables	704	(4)
11.6.2. Variance Reduction by Conditioning	708	(4)
11.6.3. Control Variates	712	(2)
11.6.4. Importance Sampling	714	(6)
11.7. Determining the Number of Runs	720	(1)
11.8. Coupling from the Past	720	(3)
Exercises	723	(8)
References	731	(2)
Appendix: Solutions to Starred Exercises	733	(42)
Index	775