Loading...
Probability models for computer science /
| Main Author: | |
|---|---|
| Format: | Printed Book |
| Published: |
New Delhi :
Elsevier,
c2002.
|
| Subjects: | |
| Online Access: | http://www.loc.gov/catdir/description/els031/2001089413.html http://www.loc.gov/catdir/toc/fy02/2001089413.html |
Table of Contents:
- Machine generated contents note: 1 Probability 1
- 1.1 Axioms of Probability 1
- 1.2 Conditional Probability and Independence 1
- 1.3 Random Variables 2
- 1.4 Expected Value and Variance 5
- 1.4.1 Expected Value and Variance of Sums of Random Variables 7
- 1.5 Moment-Generating Functions and Laplace Transforms 17
- 1.6 Conditional Expectation 20
- 1.7 Exponential Random Variables 32
- 1.8 Limit Theorems 41
- 1.8.1 Stopping Times and Wald's Equation 42
- Exercises 43
- 2 Some Examples 49
- 2.1 A Random Graph 49
- 2.2 The Quicksort and Find Algorithms 55
- 2.2.1 The Find Algorithm 58
- 2.3 A Self-Organizing List Model 61
- 2.4 Random Permutations 62
- 2.4.1 Inversions 66
- 2.4.2 Increasing Subsequences 69
- Exercises 71
- 3 Probability Bounds, Approximations,
- and Computations 75
- 3.1 'ail Probability Inequalities 75
- 3.1.1 Markov's Inequality 75
- 3.1.2 Chernoff Bounds 76
- 3.1.3 Jensen's Inequality 79
- 3.2 The Second Moment and the Conditional
- Expectation Inequality 79
- 3.3 Probability Bounds via the Importance Sampling Identity 87
- 3.4 Poisson Random Variables and the Poisson Paradigm 89
- 3.5 Compound Poisson Random Variables 94
- 3.5.1 A Second Representation when the Component
- Distribution Is Discrete 95
- 3.5.2 A Compound Poisson identity 96
- Exercises 100
- 4 Markov Chains 103
- 4.1 Introduction 103
- 4.2 Chapman-Kolmogorov Equations 105
- 4.3 Classification of States 106
- 4.4 Limiting and Stationary Probabilities 115
- 4.5 Some Applications 121
- 4.5.1 Models for Algorithmic Efficiency 121
- 4.5.2 Using a Random Walk to Analyze a Probabilistic Algorithm
- for the Satisfiability Problem 126
- 4.6 Time-Reversible Markov Chains 131
- 4.7 Markov Chain Monte Carlo Methods 142
- Exercises 147
- 5 The Probabilistic Method 151
- 5.1 Introduction 151
- 5.2 Using Probability To Prove Existence 151
- 5.3 Obtaining Bounds fiom Expectations 153
- 5.4 The Maximum Weighted Independent
- Set Problem: A Bound and a Random Algorithm 156
- 5.5 The Set-Covering Problem 161
- 5.6 Antichains 163
- 5.7 The Lovasz Local Lemma 164
- 5.8 A Random Algorithm for Finding the Minimal
- Cut in a Graph 169
- Exercises 171
- 6 Martingales 175
- 6.1 Definitions and Examples 175
- 6.2 The Martingale Stopping Theorem 177
- 6.3 The Hoeffding-Azuma Inequality 189
- 6.4 Submartingales 192
- Exercises 194
- 7 Poisson Processes 199
- 7.1 The Nonstationary Poisson Process 199
- 7.2 The Stationary Poisson Process 203
- 7.3 Some Poisson Process Computations 205
- 7.4 Classifying the Events of a Nonstationary Poisson Process 211
- 7.5 Conditional Distribution of the Arrival Times 215
- Exercises 217
- 8 Queueing Theory 221
- 8.1 Introduction 221
- 8.2 Preliminaries 222
- 8.2.1 Cost Equations 222
- 8.2.2 Steady-State Probabilities 224
- 8.3 Exponential Models 226
- 8.3.1 A Single-Server Exponential Queueing System 226
- 8.4 Birth-and-Death Exponential Queueing Systems 230
- 8.5 The Backwards Approach in Exponential Queues 238
- 8.6 A Closed Queueing Network 239
- 8.7 An Open Queueing Network 243
- 8.8 The M/G/I Queue 248
- 8.8.1 Preliminaries: Work and Another Cost Identity 248
- 8.8.2 Application of Work to M/G/1 249
- 8.8.3 Busy and idle Periods 253
- 8.8.4 Relating the Variances of Waiting
- Times and Number in System 254
- 8.9 Priority Queues 255
- Exercises 258
- 9 Simulation 261
- 9.1 Monte Carlo Simulation 261
- 9.2 Generating Discrete Random Variables 263
- 9.3 Generating Continuous Random Variables:
- The Inverse Transform Approach 266
- 9.4 The Rejection Method 268
- 9,5 Variance Reduction 272
- 9.5.1 Antithetic Variables 272
- 9.5.2 Importance Sampling 275
- 9.5.3 Variance Reduction by Conditional Expectation 279
- Exercises 280
- References 283
- Index 285.