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

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ii CONTENTS 2.12.5 Back to the Board Game Example . . . . . . . . . . . . . . . . . . . . . . . . 25 2.12.6 How Long Should We Run the Simulation? . . . . . . . . . . . . . . . . . . . 25 2.13 Combinatorics-Based Probability Computation . . . . . . . . . . . . . . . . . . . . . 26 2.13.1 Which Is More Likely in Five Cards, One King or Two Hearts? . . . . . . . . 26 2.13.2 Example: Lottery Tickets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.13.3 “Association Rules” in Data Mining . . . . . . . . . . . . . . . . . . . . . . . 28 2.13.4 Multinomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.13.5 Example: Probability of Getting Four Aces in a Bridge Hand . . . . . . . . . 30 3 Discrete Random Variables 35 3.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4.1 Generality—Not Just for DiscreteRandom Variables . . . . . . . . . . . . . . 36 3.4.1.1 What Is It? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4.3 Computation and Properties of Expected Value . . . . . . . . . . . . . . . . . 37 3.4.4 “Mailing Tubes” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.5 Casinos, Insurance Companies and “Sum Users,” Compared to Others . . . . 43 3.5 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.2 Central Importance of the Concept of Variance . . . . . . . . . . . . . . . . . 47 3.5.3 Intuition Regarding the Size of Var(X) . . . . . . . . . . . . . . . . . . . . . . 47 3.5.3.1 Chebychev’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5.3.2 The Coefficient of Variation . . . . . . . . . . . . . . . . . . . . . . . 48 3.6 Indicator Random Variables, and Their Means and Variances . . . . . . . . . . . . . 48
CONTENTS iii 3.7 A Combinatorial Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.8 A Useful Fact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.9 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.10 Expected Value, Etc. in the ALOHA Example . . . . . . . . . . . . . . . . . . . . . 53 3.11 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.11.1 Example: Toss Coin Until First Head . . . . . . . . . . . . . . . . . . . . . . 55 3.11.2 Example: Sum of Two Dice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.11.3 Example: Watts-Strogatz Random Graph Model . . . . . . . . . . . . . . . . 55 3.12 Parameteric Families of pmfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.12.1 The Geometric Family of Distributions . . . . . . . . . . . . . . . . . . . . . . 57 3.12.1.1 R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.12.1.2 Example: a Parking Space Problem . . . . . . . . . . . . . . . . . . 60 3.12.2 The Binomial Family of Distributions . . . . . . . . . . . . . . . . . . . . . . 62 3.12.2.1 R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.12.2.2 Example: Flipping Coins with Bonuses . . . . . . . . . . . . . . . . 63 3.12.2.3 Example: Analysis of Social Networks . . . . . . . . . . . . . . . . . 64 3.12.3 The Negative Binomial Family of Distributions . . . . . . . . . . . . . . . . . 65 3.12.3.1 Example: Backup Batteries . . . . . . . . . . . . . . . . . . . . . . . 66 3.12.4 The Poisson Family of Distributions . . . . . . . . . . . . . . . . . . . . . . . 66 3.12.4.1 R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.12.5 The Power Law Family of Distributions . . . . . . . . . . . . . . . . . . . . . 68 3.13 Recognizing Some Parametric Distributions When You See Them . . . . . . . . . . . 69 3.13.1 Example: a Coin Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.13.2 Example: Tossing a Set of Four Coins . . . . . . . . . . . . . . . . . . . . . . 71 3.13.3 Example: the ALOHA Example Again . . . . . . . . . . . . . . . . . . . . . . 71 3.14 Example: the Bus Ridership Problem Again . . . . . . . . . . . . . . . . . . . . . . . 72 3.15 A Cautionary Tale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Page 1 and 2: From Algorithms to Z-Scores: Probab
Page 3: Contents 1 Time Waste Versus Empowe
Page 7 and 8: CONTENTS v 5.5.1.3 Example: Modelin
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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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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