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

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2 Author’s Biographical Sketch Dr. Norm Matloff is a professor of computer science at the University of California at Davis, and was formerly a professor of statistics at that university. He is a former database software developer in Silicon Valley, and has been a statistical consultant for firms such as the Kaiser Permanente Health Plan. Dr. Matloff was born in Los Angeles, and grew up in East Los Angeles and the San Gabriel Valley. He has a PhD in pure mathematics from UCLA, specializing in probability theory and statistics. He has published numerous papers in computer science and statistics, with current research interests in parallel processing, statistical computing, and regression methodology. Prof. Matloff is a former appointed member of IFIP Working Group 11.3, an international committee concerned with database software security, established under UNESCO. He was a founding member of the UC Davis Department of Statistics, and participated in the formation of the UCD Computer Science Department as well. He is a recipient of the campuswide Distinguished Teaching Award and Distinguished Public Service Award at UC Davis. Dr. Matloff is the author of two published textbooks, and of a number of widely-used Web tutorials on computer topics, such as the Linux operating system and the Python programming language. He and Dr. Peter Salzman are authors of The Art of Debugging with GDB, DDD, and Eclipse. Prof. Matloff’s book on the R programming language, The Art of R Programming, was published in 2011. He is also the author of several open-source textbooks, including From Algorithms to Z- Scores: Probabilistic and Statistical Modeling in Computer Science (http://heather.cs.ucdavis. edu/probstatbook), and Programming on Parallel Machines (http://heather.cs.ucdavis.edu/ ~matloff/ParProcBook.pdf).
Contents 1 Time Waste Versus Empowerment 1 2 Basic Probability Models 3 2.1 ALOHA Network Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The Crucial Notion of a Repeatable Experiment . . . . . . . . . . . . . . . . . . . . 5 2.3 Our Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 “Mailing Tubes” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Basic Probability Computations: ALOHA Network Example . . . . . . . . . . . . . 10 2.6 Bayes’ Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7 ALOHA in the Notebook Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.8 Solution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.9 Example: Divisibility of Random Integers . . . . . . . . . . . . . . . . . . . . . . . . 16 2.10 Example: A Simple Board Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.11 Example: Bus Ridership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.12 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.12.1 Example: Rolling Dice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.12.2 Improving the Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.12.3 Simulation of the ALOHA Example . . . . . . . . . . . . . . . . . . . . . . . 23 2.12.4 Example: Bus Ridership, cont’d. . . . . . . . . . . . . . . . . . . . . . . . . . 24 i
Page 1: From Algorithms to Z-Scores: Probab
Page 5 and 6: CONTENTS iii 3.7 A Combinatorial Ex
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
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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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