From Algorithms to Z-Scores - matloff - University of California, Davis

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iv CONTENTS 3.15.1 Trick Coins, Tricky Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.15.2 Intuition in Retrospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.15.3 Implications for Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.16 Why Not Just Do All Analysis by Simulation? . . . . . . . . . . . . . . . . . . . . . 75 3.17 Proof of Chebychev’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.18 Reconciliation of Math and Intuition (optional section) . . . . . . . . . . . . . . . . . 77 4 Introduction to Discrete Markov Chains 83 4.1 Example: Die Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Long-Run State Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Example: 3-Heads-in-a-Row Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Example: ALOHA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5 Example: Bus Ridership Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6 An Inventory Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5 Continuous Probability Models 91 5.1 A Random Dart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Continuous Random Variables Are “Useful Unicorns” . . . . . . . . . . . . . . . . . 92 5.3 But Equation (5.2) Presents a Problem . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4 Density Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4.1 Motivation, Definition and Interpretation . . . . . . . . . . . . . . . . . . . . 96 5.4.2 Properties of Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.4.3 A First Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.5 Famous Parametric Families of Continuous Distributions . . . . . . . . . . . . . . . . 101 5.5.1 The Uniform Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.5.1.1 Density and Properties . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.5.1.2 R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
CONTENTS v 5.5.1.3 Example: Modeling of Disk Performance . . . . . . . . . . . . . . . 102 5.5.1.4 Example: Modeling of Denial-of-Service Attack . . . . . . . . . . . . 103 5.5.2 The Normal (Gaussian) Family of Continuous Distributions . . . . . . . . . . 103 5.5.2.1 Density and Properties . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5.2.2 Example: Network Intrusion . . . . . . . . . . . . . . . . . . . . . . 105 5.5.2.3 Example: Class Enrollment Size . . . . . . . . . . . . . . . . . . . . 107 5.5.2.4 More on the Jill Example . . . . . . . . . . . . . . . . . . . . . . . . 107 5.5.2.5 Example: River Levels . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.5.2.6 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 108 5.5.2.7 Example: Cumulative Roundoff Error . . . . . . . . . . . . . . . . . 109 5.5.2.8 Example: Bug Counts . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.5.2.9 Example: Coin Tosses . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.5.2.10 Museum Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5.2.11 Optional topic: Formal Statement of the CLT . . . . . . . . . . . . 111 5.5.2.12 Importance in Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5.3 The Chi-Squared Family of Distributions . . . . . . . . . . . . . . . . . . . . 112 5.5.3.1 Density and Properties . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5.3.2 Example: Error in Pin Placement . . . . . . . . . . . . . . . . . . . 113 5.5.3.3 Importance in Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.4 The Exponential Family of Distributions . . . . . . . . . . . . . . . . . . . . . 114 5.5.4.1 Density and Properties . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.4.2 R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.4.3 Example: Refunds on Failed Components . . . . . . . . . . . . . . . 115 5.5.4.4 Example: Overtime Parking Fees . . . . . . . . . . . . . . . . . . . . 115 5.5.4.5 Connection to the Poisson Distribution Family . . . . . . . . . . . . 116 5.5.4.6 Importance in Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.5.5 The Gamma Family of Distributions . . . . . . . . . . . . . . . . . . . . . . . 118
Page 1 and 2: From Algorithms to Z-Scores: Probab
Page 3 and 4: Contents 1 Time Waste Versus Empowe
Page 5: CONTENTS iii 3.7 A Combinatorial Ex
Page 9 and 10: CONTENTS vii 7.3.1 Properties of Me
Page 11 and 12: CONTENTS ix 10.1 Sampling Distribut
Page 13 and 14: CONTENTS xi 11.9.4 What to Do Inste
Page 15 and 16: CONTENTS xiii 15.2 Example Applicat
Page 17 and 18: CONTENTS xv 17.2.3 Logistic Regress
Page 19 and 20: CONTENTS xvii 19.2 Simulation of Ra
Page 21 and 22: CONTENTS xix 21.4 Loss Models . . .
Page 23 and 24: Preface Why is this book different
Page 25 and 26: Chapter 1 Time Waste Versus Empower
Page 27 and 28: Chapter 2 Basic Probability Models
Page 29 and 30: 2.2. THE CRUCIAL NOTION OF A REPEAT
Page 31 and 32: 2.3. OUR DEFINITIONS 7 2009, cannot
Page 33 and 34: 2.4. “MAILING TUBES” 9 but in m
Page 35 and 36: 2.5. BASIC PROBABILITY COMPUTATIONS
Page 37 and 38: 2.6. BAYES’ RULE 13 Note by the w
Page 39 and 40: 2.8. SOLUTION STRATEGIES 15 2.8 Sol
Page 41 and 42: 2.10. EXAMPLE: A SIMPLE BOARD GAME
Page 43 and 44: 2.11. EXAMPLE: BUS RIDERSHIP 19 Aga
Page 45 and 46: 2.12. SIMULATION 21 1 # roll d dice
Page 47 and 48: 2.12. SIMULATION 23 So, in evaluati
Page 49 and 50: 2.12. SIMULATION 25 3 count
Page 51 and 52: 2.13. COMBINATORICS-BASED PROBABILI
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2.13. COMBINATORICS-BASED PROBABILI
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Chapter 3 Discrete Random Variables
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3.4. EXPECTED VALUE 37 3.4.1.1 What
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3.4. EXPECTED VALUE 39 So It turns
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3.4. EXPECTED VALUE 41 • For rand
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3.4. EXPECTED VALUE 43 of two rando
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3.5. VARIANCE 45 ance of U is defin
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3.5. VARIANCE 47 for any constant d
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3.7. A COMBINATORIAL EXAMPLE 49 You
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3.8. A USEFUL FACT 51 Note carefull
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3.10. EXPECTED VALUE, ETC. IN THE A
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3.11. DISTRIBUTIONS 55 3.11.1 Examp
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3.12. PARAMETERIC FAMILIES OF PMFS
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3.13. RECOGNIZING SOME PARAMETRIC D
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3.13. RECOGNIZING SOME PARAMETRIC D
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3.15. A CAUTIONARY TALE 73 T has a
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3.16. WHY NOT JUST DO ALL ANALYSIS
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3.18. RECONCILIATION OF MATH AND IN
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Chapter 4 Introduction to Discrete
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4.3. EXAMPLE: 3-HEADS-IN-A-ROW GAME
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4.4. EXAMPLE: ALOHA 87 The quantity
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4.6. AN INVENTORY MODEL 89 4.6 An I
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Chapter 5 Continuous Probability Mo
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5.3. BUT EQUATION (??) PRESENTS A P
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5.3. BUT EQUATION (??) PRESENTS A P
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5.4. DENSITY FUNCTIONS 97 2(0.1)fX(
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5.4. DENSITY FUNCTIONS 99 5.4.2 Pro
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5.5. FAMOUS PARAMETRIC FAMILIES OF
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5.8. “HYBRID” CONTINUOUS/DISCRE
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5.8. “HYBRID” CONTINUOUS/DISCRE
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Chapter 6 Stop and Review: Probabil
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• famous parametric families of d
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Chapter 7 Covariance and Random Vec
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7.1. MEASURING CO-VARIATION OF RAND
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7.2. SETS OF INDEPENDENT RANDOM VAR
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7.2. SETS OF INDEPENDENT RANDOM VAR
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7.3. MATRIX FORMULATIONS 139 this l
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7.3. MATRIX FORMULATIONS 141 consis
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7.3. MATRIX FORMULATIONS 143 import
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7.3. MATRIX FORMULATIONS 145 since
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7.3. MATRIX FORMULATIONS 147 minimi
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Chapter 8 Multivariate PMFs and Den
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8.1. MULTIVARIATE PROBABILITY MASS
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8.2. MULTIVARIATE DENSITIES 153 So,
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8.2. MULTIVARIATE DENSITIES 155 we
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8.3. MORE ON SETS OF INDEPENDENT RA
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8.4. EXAMPLE: FINDING THE DISTRIBUT
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8.5. PARAMETRIC FAMILIES OF MULTIVA
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Chapter 9 Introduction to Continuou
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9.1. MEMORYLESS PROPERTY OF EXPONEN
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9.3. HOLDING-TIME DISTRIBUTION 187
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Chapter 10 Introduction to Confiden
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10.1. SAMPLING DISTRIBUTIONS 195 Wh
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10.1. SAMPLING DISTRIBUTIONS 197 Ap
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10.3. CONFIDENCE INTERVALS FOR MEAN
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10.4. MEANING OF CONFIDENCE INTERVA
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10.5. GENERAL FORMATION OF CONFIDEN
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10.7. CONFIDENCE INTERVALS FOR PROP
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10.7. CONFIDENCE INTERVALS FOR PROP
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10.8. CONFIDENCE INTERVALS FOR DIFF
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10.10. R COMPUTATION 215 algebra, w
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10.13. OTHER CONFIDENCE LEVELS 217
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10.14. ONE MORE TIME: WHY DO WE USE
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Chapter 11 Introduction to Signific
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11.2. GENERAL TESTING BASED ON NORM
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11.5. ONE-SIDED HA 225 By checking
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11.6. EXACT TESTS 227 It is natural
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11.8. THE POWER OF A TEST 229 11.8
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11.9. WHAT’S WRONG WITH SIGNIFICA
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Chapter 12 General Statistical Esti
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12.1. GENERAL METHODS OF PARAMETRIC
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12.2. BIAS AND VARIANCE 247 people,
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12.2. BIAS AND VARIANCE 249 Moreove
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12.3. MORE ON THE ISSUE OF INDEPEND
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12.4. NONPARAMETRIC DISTRIBUTION ES
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12.5. BAYESIAN METHODS 259 his plan
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12.5. BAYESIAN METHODS 261 it now b
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12.5. BAYESIAN METHODS 263 number
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12.5. BAYESIAN METHODS 265 (b) ˆp,
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Chapter 13 Simultaneous Inference M
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13.2. SCHEFFE’S METHOD 269 You ca
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13.4. OTHER METHODS FOR SIMULTANEOU
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Chapter 14 Introduction to Model Bu
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14.1. “DESPERATE FOR DATA” 275
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14.1. “DESPERATE FOR DATA” 277
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14.2. ASSESSING “GOODNESS OF FIT
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14.4. ROBUSTNESS 281 bin width, or
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14.5. REAL POPULATIONS AND CONCEPTU
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Chapter 15 Relations Among Variable
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15.3. ADJUSTING FOR COVARIATES 287
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15.6. ESTIMATING THAT RELATIONSHIP
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15.6. ESTIMATING THAT RELATIONSHIP
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15.7. EXAMPLE: BASEBALL DATA 293
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15.9. EXAMPLE: BASEBALL DATA (CONT
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15.11. PREDICTION 297 matter, it ma
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15.12. PARAMETRIC ESTIMATION OF LIN
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15.12. PARAMETRIC ESTIMATION OF LIN
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15.14. DUMMY VARIABLES 303 15.14 Du
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15.16. WHAT DOES IT ALL MEAN?—EFF
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15.17. MODEL SELECTION 307 But look
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15.17. MODEL SELECTION 309 Foundati
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15.18. WHAT ABOUT THE ASSUMPTIONS?
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15.19. CASE STUDIES 313 have an und
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15.19. CASE STUDIES 315 Exercises N
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15.19. CASE STUDIES 317 8. Consider
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Chapter 16 Advanced Statistical Est
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16.2. THE DELTA METHOD: CONFIDENCE
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16.3. THE BOOTSTRAP METHOD FOR FORM
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16.3. THE BOOTSTRAP METHOD FOR FORM
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Chapter 17 Relations Among Variable
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17.2. THE CLASSIFICATION PROBLEM 33
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17.3. NONPARAMETRIC ESTIMATION OF R
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17.4. SYMMETRIC RELATIONS AMONG SEV
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17.5. SIMPSON’S (NON-)PARADOX 357
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17.5. SIMPSON’S (NON-)PARADOX 359
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Chapter 18 Describing “Failure”
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18.2. A CAUTIONARY TALE: THE BUS PA
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18.2. A CAUTIONARY TALE: THE BUS PA
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18.3. RESIDUAL-LIFE DISTRIBUTION 36
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Chapter 19 Advanced Multivariate Me
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19.1. CONDITIONAL DISTRIBUTIONS 377
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19.3. MIXTURE MODELS 383 Now that w
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19.4. TRANSFORM METHODS 385 Thus an
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19.4. TRANSFORM METHODS 387 19.4.2
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19.4. TRANSFORM METHODS 389 transfo
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19.5. VECTOR SPACE INTERPRETATIONS
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19.7. CONDITIONAL EXPECTATION AS A
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19.8. PROOF OF THE LAW OF TOTAL EXP
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19.8. PROOF OF THE LAW OF TOTAL EXP
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Chapter 20 Markov Chains One of the
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20.1. DISCRETE-TIME MARKOV CHAINS 4
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20.2. SIMULATION OF MARKOV CHAINS 4
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20.4. CONTINUOUS-TIME MARKOV CHAINS
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20.5. HITTING TIMES ETC. 419 least
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20.5. HITTING TIMES ETC. 421 First
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20.5. HITTING TIMES ETC. 423 valued
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20.5. HITTING TIMES ETC. 425 summon
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Chapter 21 Introduction to Queuing
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21.2. M/M/1 429 busy for approximat
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21.2. M/M/1 431 • Due to the memo
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21.3. MULTI-SERVER MODELS 433 Recal
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21.4. LOSS MODELS 435 1 = i,j,k π
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21.5. NONEXPONENTIAL SERVICE TIMES
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21.6. REVERSED MARKOV CHAINS 439 So
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21.6. REVERSED MARKOV CHAINS 441 21
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21.6. REVERSED MARKOV CHAINS 443 Re
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21.7. NETWORKS OF QUEUES 445 • Gi
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21.7. NETWORKS OF QUEUES 447 Let Li
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Appendix A Review of Matrix Algebra
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A.2. MATRIX TRANSPOSE 451 • Matri
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A.6. EIGENVALUES AND EIGENVECTORS 4
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Appendix B R Quick Start Here we pr
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B.3. FIRST SAMPLE PROGRAMMING SESSI
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B.4. SECOND SAMPLE PROGRAMMING SESS
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B.4. SECOND SAMPLE PROGRAMMING SESS
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B.6. COMPLEX NUMBERS 463 B.6 Comple
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