Introduction to probability models / Sheldon M. Ross.
Editor: Amsterdam : Academic Press, c2007Edición: 9th edDescripción: xviii, 782 p. : il. ; 24 cmISBN: 9780125980623 (acidfree paper); 0125980620 (acid-free paper); 9780123736352 (pbk. : acid-free paper); 0123736358 (pbk. : acid-free paper)Tema(s): ProbabilitiesOtra clasificación: 60-01 (62-01) Recursos en línea: Publisher description1. Introduction to Probability Theory [1] LI. Introduction [1] 1.2. Sample Space and Events [1] 1.3. Probabilities Defined on Events [4] 1.4. Conditional Probabilities [7] 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 [28] 2.2.2. The Binomial Random Variable [29] 2.2.3. The Geometric Random Variable [31] 2.2.4. The Poisson Random Variable [32] 2.3. Continuous Random Variables [34] 2.3.1. The Uniform Random Variable [35] 2.3.2. Exponential Random Variables [36] 2.3.3. Gamma Random Variables [37] 2.3.4. Normal Random Variables [37] 2.4. Expectation of a Random Variable [38] 2.4.1. The Discrete Case [38] 2.4.2. The Continuous Case [41] 2.4.3. Expectation of a Function of a Random Variable [43] 2.5. Jointly Distributed Random Variables [47] 2.5.1. Joint Distribution Functions [47] 2.5.2. Independent Random Variables [51] 2.5.3. Covariance and Variance of Sums of Random Variables [53] 2.5.4. Joint Probability Distribution of Functions of Random Variables [61] 2.6. Moment Generating Functions [64] 2.6.1. The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population [74] 2.7. Limit Theorems [77] 2.8. Stochastic Processes [83] Exercises [85] References [96] 3. Conditional Probability and Conditional Expectation [97] 3.1. Introduction [97] 3.2. The Discrete Case [97] 3.3. The Continuous Case [102] 3.4. Computing Expectations by Conditioning [105] 3.4.1. Computing Variances by Conditioning [117] 3.5. Computing Probabilities by Conditioning [120] 3.6. Some Applications [137] 3.6.1. A List Model [137] 3.6.2. A Random Graph [139] 3.6.3. Uniform Priors, Polya’s Um Model, and Bose-Einstein Statistics [147] 3.6.4. Mean Time for Patterns [151] 3.6.5. The ^-Record Values of Discrete Random Variables [155] 3.7. An Identity for Compound Random Variables [158] 3.7.1. Poisson Compounding Distribution [161] 3.7.2. Binomial Compounding Distribution [163] 3.7.3. A Compounding Distribution Related to the Negative Binomial [164] Exercises [165] 4. Markov Chains [185] 4.1. Introduction [185] 4.2. Chapman-Kolmogorov Equations [189] 4.3. Classification of States [193] 4.4. Limiting Probabilities [204] 4.5. Some Applications [217] 4.5.1. The Gambler’s Ruin Problem [217] 4.5.2. A Model for Algorithmic Efficiency [221] 4.5.3. Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem [224] 4.6. Mean Time Spent in Transient States [230] 4.7. Branching Processes [233] 4.8. Time Reversible Markov Chains [236] 4.9. Markov Chain Monte Carlo Methods [247] 4.10. Markov Decision Processes [252] 4.11. Hidden Markov Chains [256] 4.11.1. Predicting the States [261] Exercises [263] References [280] 5. The Exponential Distribution and the Poisson Process [281] 5.1. Introduction [281] 5.2. The Exponential Distribution [282] 5.2.1. Definition [282] 5.2.2. Properties of the Exponential Distribution [284] 5.2.3. Further Properties of the Exponential Distribution [291] 5.2.4. Convolutions of Exponential Random Variables [298] 5.3. The Poisson Process [302] 5.3.1. Counting Processes [302] 5.3.2. Definition of the Poisson Process [304] 5.3.3. Interarrival and Waiting Time Distributions [307] 5.3.4. Further Properties of Poisson Processes [310] 5.3.5. Conditional Distribution of the Arrival Times [316] 5.3.6. Estimating Software Reliability [328] 5.4. Generalizations of the Poisson Process [330] 5.4.1. Nonhomogeneous Poisson Process [330] 5.4.2. Compound Poisson Process [337] 5.4.3. Conditional or Mixed Poisson Processes [343] Exercises [346] References [364] 6. Continuous-Time Markov Chains [365] 6.1. Introduction [365] 6.2. Continuous-Time Markov Chains [366] 6.3. Birth and Death Processes [368] 6.4. The Transition Probability Function Pij(f) [375] 6.5. Limiting Probabilities [384] 6.6. Time Reversibility [392] 6.7. Uniformization [401] 6.8. Computing the Transition Probabilities [404] Exercises [407] References [415] 7. Renewal Theory and Its Applications [417] 7.1. Introduction [417] 7.2. Distribution of N(f) [419] 7.3. Limit Theorems and Their Applications [423] 7.4. Renewal Reward Processes [433] 7.5. Regenerative Processes [442] 7.5.1. Alternating Renewal Processes [445] 7.6. Semi-Markov Processes [452] 7.7. The Inspection Paradox [455] 7.8. Computing the Renewal Function [458] 7.9. Applications to Patterns [461] 7.9.1. Patterns of Discrete Random Variables [462] 7.9.2. The Expected Time to a Maximal Run of Distinct Values [469] 7.9.3. Increasing Runs of Continuous Random Variables [471] 7.10. The Insurance Ruin Problem [473] Exercises [479] References [492] 8. Queueing Theory [493] 8.1. Introduction [493] 8.2. Preliminaries [494] 8.2.1. Cost Equations [495] 8.2.2. Steady-State Probabilities [496] 8.3. Exponential Models [499] 8.3.1. A Single-Server Exponential Queueing System [499] 8.3.2. A Single-Server Exponential Queueing System Having Finite Capacity [508] 8.3.3. A Shoeshine Shop [511] 8.3.4. A Queueing System with Bulk Service [514] 8.4. Network of Queues [517] 8.4.1. Open Systems [517] 8.4.2. Closed Systems [522] 8.5. The System M/G/1 [528] 8.5.1. Preliminaries: Work and Another Cost Identity [528] 8.5.2. Application of Work to M/G/1 [529] 8.5.3. Busy Periods [530] 8.6. Variations on the M/G/1 [531] 8.6.1. The M/G/1 with Random-Sized Batch Arrivals [531] 8.6.2. Priority Queues [533] 8.6.3. An M/G/l Optimization Example [536] 8.6.4. The M/G/1 Queue with Server Breakdown [540] 8.7. The Model G/M/1 [543] 8.7.1. The G/M/X Busy and Idle Periods [548] 8.8. A Finite Source Model [549] 8.9. Multiserver Queues [552] 8.9.1. Erlang’s Loss System [553] 8.9.2. The M/M/k Queue [554] 8.9.3. The G/M/k Queue [554] 8.9.4. The M/G/k Queue [556] Exercises [558] References [570] 9. Reliability Theory [571] 9.1. Introduction [571] 9.2. Structure Functions [571] 9.2.1. Minimal Path and Minimal Cut Sets [574] 9.3. Reliability of Systems of Independent Components [578] 9.4. Bounds on the Reliability Function [583] 9.4.1. Method of Inclusion and Exclusion [584] 9.4.2. Second Method for Obtaining Bounds on r(p) [593] 9.5. System Life as a Function of Component Lives [595] 9.6. Expected System Lifetime [604] 9.6.1. An Upper Bound on the Expected Life of a Parallel System [608] 9.7. Systems with Repair [610] 9.7.1. A Series Model with Suspended Animation 615 Exercises [617] References [624] 10. Brownian Motion and Stationary Processes [625] 10.1. Brownian Motion [625] 10.2. Hitting Times, Maximum Variable, and the Gambler’s Ruin Problem [629] 10.3. Variations on Brownian Motion [631] 10.3.1. Brownian Motion with Drift [631] 10.3.2. Geometric Brownian Motion [631] 10.4. Pricing Stock Options [632] 10.4.1. An Example in Options Pricing [632] 10.4.2. The Arbitrage Theorem [635] 10.4.3. The Black-Scholes Option Pricing Formula [638] 10.5. White Noise [644] 10.6. Gaussian Processes [646] 10.7. Stationary and Weakly Stationary Processes [649] 10.8. Harmonic Analysis of Weakly Stationary Processes [654] Exercises [657] References [662] 11. Simulation [663] 11.1. Introduction [663] 11.2. General Techniques for Simulating Continuous Random Variables [668] 11.2.1. The Inverse Transformation Method [668] 11.2.2. The Rejection Method [669] 11.2.3. The Hazard Rate Method [673] 11.3. Special Techniques for Simulating Continuous Random Variables [677] 11.3.1. The Normal Distribution [677] 11.3.2. The Gamma Distribution [680] 11.3.3. The Chi-Squared Distribution [681] 11.3.4. The Beta (n, m) Distribution [681] 11.3.5. The Exponential Distribution—The Von Neumann Algorithm [682] 11.4. Simulating from Discrete Distributions [685] 11.4.1. The Alias Method [688] 11.5. Stochastic Processes [692] 11.5.1. Simulating a Nonhomogeneous Poisson Process [693] 11.5.2. Simulating a Two-Dimensional Poisson Process [700] 11.6. Variance Reduction Techniques [703] 11.6.1. Use of Antithetic Variables [704] 11.6.2. Variance Reduction by Conditioning [708] 11.6.3. Control Variates [712] 11.6.4. Importance Sampling [714] 11.7. Determining the Number of Runs [720] 11.8. Coupling from the Past [720] Exercises [723] References [731] Appendix: Solutions to Starred Exercises [733] Index [775]
Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Libros | Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 60 R823 (Browse shelf) | Available | A-8552 |
Incluye referencias bibliográficas e índice.
1. Introduction to Probability Theory [1] --
LI. Introduction [1] --
1.2. Sample Space and Events [1] --
1.3. Probabilities Defined on Events [4] --
1.4. Conditional Probabilities [7] --
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 [28] --
2.2.2. The Binomial Random Variable [29] --
2.2.3. The Geometric Random Variable [31] --
2.2.4. The Poisson Random Variable [32] --
2.3. Continuous Random Variables [34] --
2.3.1. The Uniform Random Variable [35] --
2.3.2. Exponential Random Variables [36] --
2.3.3. Gamma Random Variables [37] --
2.3.4. Normal Random Variables [37] --
2.4. Expectation of a Random Variable [38] --
2.4.1. The Discrete Case [38] --
2.4.2. The Continuous Case [41] --
2.4.3. Expectation of a Function of a Random Variable [43] --
2.5. Jointly Distributed Random Variables [47] --
2.5.1. Joint Distribution Functions [47] --
2.5.2. Independent Random Variables [51] --
2.5.3. Covariance and Variance of Sums of Random Variables [53] --
2.5.4. Joint Probability Distribution of Functions of Random Variables [61] --
2.6. Moment Generating Functions [64] --
2.6.1. The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population [74] --
2.7. Limit Theorems [77] --
2.8. Stochastic Processes [83] --
Exercises [85] --
References [96] --
3. Conditional Probability and Conditional Expectation [97] --
3.1. Introduction [97] --
3.2. The Discrete Case [97] --
3.3. The Continuous Case [102] --
3.4. Computing Expectations by Conditioning [105] --
3.4.1. Computing Variances by Conditioning [117] --
3.5. Computing Probabilities by Conditioning [120] --
3.6. Some Applications [137] --
3.6.1. A List Model [137] --
3.6.2. A Random Graph [139] --
3.6.3. Uniform Priors, Polya’s Um Model, and Bose-Einstein Statistics [147] --
3.6.4. Mean Time for Patterns [151] --
3.6.5. The ^-Record Values of Discrete Random Variables [155] --
3.7. An Identity for Compound Random Variables [158] --
3.7.1. Poisson Compounding Distribution [161] --
3.7.2. Binomial Compounding Distribution [163] --
3.7.3. A Compounding Distribution Related to the Negative Binomial [164] --
Exercises [165] --
4. Markov Chains [185] --
4.1. Introduction [185] --
4.2. Chapman-Kolmogorov Equations [189] --
4.3. Classification of States [193] --
4.4. Limiting Probabilities [204] --
4.5. Some Applications [217] --
4.5.1. The Gambler’s Ruin Problem [217] --
4.5.2. A Model for Algorithmic Efficiency [221] --
4.5.3. Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem [224] --
4.6. Mean Time Spent in Transient States [230] --
4.7. Branching Processes [233] --
4.8. Time Reversible Markov Chains [236] --
4.9. Markov Chain Monte Carlo Methods [247] --
4.10. Markov Decision Processes [252] --
4.11. Hidden Markov Chains [256] --
4.11.1. Predicting the States [261] --
Exercises [263] --
References [280] --
5. The Exponential Distribution and the Poisson Process [281] --
5.1. Introduction [281] --
5.2. The Exponential Distribution [282] --
5.2.1. Definition [282] --
5.2.2. Properties of the Exponential Distribution [284] --
5.2.3. Further Properties of the Exponential Distribution [291] --
5.2.4. Convolutions of Exponential Random Variables [298] --
5.3. The Poisson Process [302] --
5.3.1. Counting Processes [302] --
5.3.2. Definition of the Poisson Process [304] --
5.3.3. Interarrival and Waiting Time Distributions [307] --
5.3.4. Further Properties of Poisson Processes [310] --
5.3.5. Conditional Distribution of the Arrival Times [316] --
5.3.6. Estimating Software Reliability [328] --
5.4. Generalizations of the Poisson Process [330] --
5.4.1. Nonhomogeneous Poisson Process [330] --
5.4.2. Compound Poisson Process [337] --
5.4.3. Conditional or Mixed Poisson Processes [343] --
Exercises [346] --
References [364] --
6. Continuous-Time Markov Chains [365] --
6.1. Introduction [365] --
6.2. Continuous-Time Markov Chains [366] --
6.3. Birth and Death Processes [368] --
6.4. The Transition Probability Function Pij(f) [375] --
6.5. Limiting Probabilities [384] --
6.6. Time Reversibility [392] --
6.7. Uniformization [401] --
6.8. Computing the Transition Probabilities [404] --
Exercises [407] --
References [415] --
7. Renewal Theory and Its Applications [417] --
7.1. Introduction [417] --
7.2. Distribution of N(f) [419] --
7.3. Limit Theorems and Their Applications [423] --
7.4. Renewal Reward Processes [433] --
7.5. Regenerative Processes [442] --
7.5.1. Alternating Renewal Processes [445] --
7.6. Semi-Markov Processes [452] --
7.7. The Inspection Paradox [455] --
7.8. Computing the Renewal Function [458] --
7.9. Applications to Patterns [461] --
7.9.1. Patterns of Discrete Random Variables [462] --
7.9.2. The Expected Time to a Maximal Run of Distinct Values [469] --
7.9.3. Increasing Runs of Continuous Random Variables [471] --
7.10. The Insurance Ruin Problem [473] --
Exercises [479] --
References [492] --
8. Queueing Theory [493] --
8.1. Introduction [493] --
8.2. Preliminaries [494] --
8.2.1. Cost Equations [495] --
8.2.2. Steady-State Probabilities [496] --
8.3. Exponential Models [499] --
8.3.1. A Single-Server Exponential Queueing System [499] --
8.3.2. A Single-Server Exponential Queueing System Having Finite Capacity [508] --
8.3.3. A Shoeshine Shop [511] --
8.3.4. A Queueing System with Bulk Service [514] --
8.4. Network of Queues [517] --
8.4.1. Open Systems [517] --
8.4.2. Closed Systems [522] --
8.5. The System M/G/1 [528] --
8.5.1. Preliminaries: Work and Another Cost Identity [528] --
8.5.2. Application of Work to M/G/1 [529] --
8.5.3. Busy Periods [530] --
8.6. Variations on the M/G/1 [531] --
8.6.1. The M/G/1 with Random-Sized Batch Arrivals [531] --
8.6.2. Priority Queues [533] --
8.6.3. An M/G/l Optimization Example [536] --
8.6.4. The M/G/1 Queue with Server Breakdown [540] --
8.7. The Model G/M/1 [543] --
8.7.1. The G/M/X Busy and Idle Periods [548] --
8.8. A Finite Source Model [549] --
8.9. Multiserver Queues [552] --
8.9.1. Erlang’s Loss System [553] --
8.9.2. The M/M/k Queue [554] --
8.9.3. The G/M/k Queue [554] --
8.9.4. The M/G/k Queue [556] --
Exercises [558] --
References [570] --
9. Reliability Theory [571] --
9.1. Introduction [571] --
9.2. Structure Functions [571] --
9.2.1. Minimal Path and Minimal Cut Sets [574] --
9.3. Reliability of Systems of Independent Components [578] --
9.4. Bounds on the Reliability Function [583] --
9.4.1. Method of Inclusion and Exclusion [584] --
9.4.2. Second Method for Obtaining Bounds on r(p) [593] --
9.5. System Life as a Function of Component Lives [595] --
9.6. Expected System Lifetime [604] --
9.6.1. An Upper Bound on the Expected Life of a Parallel System [608] --
9.7. Systems with Repair [610] --
9.7.1. A Series Model with Suspended Animation 615 Exercises [617] --
References [624] --
10. Brownian Motion and Stationary Processes [625] --
10.1. Brownian Motion [625] --
10.2. Hitting Times, Maximum Variable, and the Gambler’s Ruin Problem [629] --
10.3. Variations on Brownian Motion [631] --
10.3.1. Brownian Motion with Drift [631] --
10.3.2. Geometric Brownian Motion [631] --
10.4. Pricing Stock Options [632] --
10.4.1. An Example in Options Pricing [632] --
10.4.2. The Arbitrage Theorem [635] --
10.4.3. The Black-Scholes Option Pricing Formula [638] --
10.5. White Noise [644] --
10.6. Gaussian Processes [646] --
10.7. Stationary and Weakly Stationary Processes [649] --
10.8. Harmonic Analysis of Weakly Stationary Processes [654] --
Exercises [657] --
References [662] --
11. Simulation [663] --
11.1. Introduction [663] --
11.2. General Techniques for Simulating Continuous Random Variables [668] --
11.2.1. The Inverse Transformation Method [668] --
11.2.2. The Rejection Method [669] --
11.2.3. The Hazard Rate Method [673] --
11.3. Special Techniques for Simulating Continuous Random Variables [677] --
11.3.1. The Normal Distribution [677] --
11.3.2. The Gamma Distribution [680] --
11.3.3. The Chi-Squared Distribution [681] --
11.3.4. The Beta (n, m) Distribution [681] --
11.3.5. The Exponential Distribution—The Von Neumann Algorithm [682] --
11.4. Simulating from Discrete Distributions [685] --
11.4.1. The Alias Method [688] --
11.5. Stochastic Processes [692] --
11.5.1. Simulating a Nonhomogeneous Poisson Process [693] --
11.5.2. Simulating a Two-Dimensional Poisson Process [700] --
11.6. Variance Reduction Techniques [703] --
11.6.1. Use of Antithetic Variables [704] --
11.6.2. Variance Reduction by Conditioning [708] --
11.6.3. Control Variates [712] --
11.6.4. Importance Sampling [714] --
11.7. Determining the Number of Runs [720] --
11.8. Coupling from the Past [720] --
Exercises [723] --
References [731] --
Appendix: Solutions to Starred Exercises [733] --
Index [775] --
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