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

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xx CONTENTS A.5 Matrix Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 A.6 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 B R Quick Start 455 B.1 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 B.2 Starting R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 B.3 First Sample Programming Session . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 B.4 Second Sample Programming Session . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 B.5 Third Sample Programming Session . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 B.6 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 B.7 Other Sources for Learning R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 B.8 Online Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 B.9 Debugging in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
Preface Why is this book different from all other books on mathematical probability and statistics, especially those for computer science students? What sets this text apart is its consistently applied approach, in a number of senses: First, there is a strong emphasis on intution, with less mathematical formalism. In my experience, defining probability via sample spaces, the standard approach, is a major impediment to doing good applied work. The same holds for defining expected value as a weighted average. Instead, I use the intuitive, informal approach of long-run frequency and long-run average. However, in spite of the relative lack of formalism, all models and so on are described precisely in terms of random variables and distributions. And the material is actually somewhat more mathematical than most at this level in the sense that it makes extensive usage of linear algebra. Second, the book stresses real-world applications. Many similar texts, notably the elegant and interesting book by Mitzenmacher, focus on probability, in fact discrete probability. Their intended class of “applications” is the theoretical analysis of algorithms. I instead focus on the actual use of the material in the real world; which tends to be more continuous than discrete, and more in the realm of statistics than probability. This should prove especially valuable, as “big data” and machine learning now play a significant role in applications of computers. Third, there is a strong emphasis on modeling. Considerable emphasis is placed on questions such as: What do probabilistic models really mean, in real-life terms? How does one choose a model? How do we assess the practical usefulness of models? This aspect is so important that there is a separate chapter for this, titled Introduction to Model Building. Throughout the text, there is considerable discussion of the real-world meaning of probabilistic concepts. For instance, when probability density functions are introduced, there is an extended discussion regarding the intuitive meaning of densities in light of the inherently-discrete nature of real data, due to the finite precision of measurement. Finally, the R statistical/data manipulation language is used throughout. Again, several excellent texts on probability and statistics have been written that feature R, but this book, by virtue of having a computer science audience, uses R in a more sophisticated manner. My open source xxi
Page 1 and 2: From Algorithms to Z-Scores: Probab
Page 3 and 4: Contents 1 Time Waste Versus Empowe
Page 5 and 6: CONTENTS iii 3.7 A Combinatorial Ex
Page 7 and 8: CONTENTS v 5.5.1.3 Example: Modelin
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Page 11 and 12: CONTENTS ix 10.1 Sampling Distribut
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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: CONTENTS xix 21.4 Loss Models . . .
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
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Page 37 and 38: 2.6. BAYES’ RULE 13 Note by the w
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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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Page 59 and 60: Chapter 3 Discrete Random Variables
Page 61 and 62: 3.4. EXPECTED VALUE 37 3.4.1.1 What
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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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