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164 CHAPTER 8. MULTIVARIATE PMFS AND DENSITIES Scale time so that for hosts A and B above, the mean time to infection is 1.0. Since in state i there are i(g-i) such pairs, the time to the next state transition is the minimum of i(g-i) exponentiallydistributed random variables with mean 1. Thus the mean time to go from state i to state i+1 is 1/[i(g-i)]. Then the mean time to go from state 1 to state g-1 is Using a calculus approximation, we have g−1 1 1 x(g − x) dx = 1 g g−1 i=1 g−1 1 1 i(g − i) ( 1 x (8.84) 1 2 + ) dx = ln(g − 1) (8.85) g − x g The latter quantity goes to zero as g → ∞. This confirms that the behavior seen by the student in simulations holds in general. In other words, (8.84) remains bounded as g → ∞. This is a very interesting result, since it says that the mean time to alert is bounded no matter how big our group size is. So, even though our model here was quite simple, probably overly so, it did explain why the student was seeing the surprising behavior in his simulations. 8.3.9 Example: Electronic Components Suppose we have three electronic parts, with independent lifetimes that are exponentially distributed with mean 2.5. They are installed simultaneously. Let’s find the mean time until the last failure occurs. Actually, we can use the same reasoning as for the computer worm example in Section 8.3.8: The mean time is simply 8.3.10 Example: Ethernet Again 1/(3 · 0.4) + 1/(2 · 0.4) + 1/(1 · 0.4) (8.86) In the Ethernet example in Section 8.3.3, we assumed that transmission time was a constant, 0.1. Now let’s account for messages of varying sizes, by assuming that transmission time T for a message is random, exponentially distributed with mean 0.1. Let’s find P (X < Y and there is no collision).
8.4. EXAMPLE: FINDING THE DISTRIBUTION OF THE SUM OF NONINDEPENDENT RANDOM VARIAB That probability is equal to P (X + T < Y ). Well, this sounds like we’re going to have to deal with triple integrals, but actually not. The derivation in Section 8.3.5 shows that the density of S = X+T is Thus the joint density of S and Y is fS(t) = e −0.1t − e −0.2t , 0 < t < ∞ (8.87) fS,Y (u, v) = (e −0.1u − e −0.2u )0.2e −0.2v , 0 < u, v, < ∞ (8.88) We can then evaluate P (S < Y ) as a double integral, along the same lines as we did for instance in (8.19). 8.4 Example: Finding the Distribution of the Sum of Nonindependent Random Variables In Section 8.3.2, we found a general formula for the distribution of the sum of two independent random variables. What about the nonindependent case? Suppose for instance fX,Y (s, t) = 2 on 0 < t < s < 1, 0 elsewhere. Let’s find fX+Y (w) for the case 0 < w < 1. Since X and Y are not independent, we cannot use convolution. But: So fX+Y (w) = w. The case 1 < w < 2 is similar. FX+Y (w) = P (X + Y ) ≤ w (8.89) = w/2 w−t 0 t 2 ds dt (8.90) = w 2 /2 (8.91) 8.5 Parametric Families of Multivariate Distributions Since there are so many ways in which random variables can correlate with each other, there are rather few parametric families commonly used to model multivariate distributions (other than
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From Algorithms to Z-Scores: Probab
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Contents 1 Time Waste Versus Empowe
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CONTENTS iii 3.7 A Combinatorial Ex
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CONTENTS v 5.5.1.3 Example: Modelin
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CONTENTS vii 7.3.1 Properties of Me
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CONTENTS ix 10.1 Sampling Distribut
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CONTENTS xi 11.9.4 What to Do Inste
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CONTENTS xiii 15.2 Example Applicat
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CONTENTS xv 17.2.3 Logistic Regress
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CONTENTS xvii 19.2 Simulation of Ra
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CONTENTS xix 21.4 Loss Models . . .
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Preface Why is this book different
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Chapter 1 Time Waste Versus Empower
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Chapter 2 Basic Probability Models
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2.2. THE CRUCIAL NOTION OF A REPEAT
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2.3. OUR DEFINITIONS 7 2009, cannot
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2.4. “MAILING TUBES” 9 but in m
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2.5. BASIC PROBABILITY COMPUTATIONS
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2.6. BAYES’ RULE 13 Note by the w
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2.8. SOLUTION STRATEGIES 15 2.8 Sol
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2.10. EXAMPLE: A SIMPLE BOARD GAME
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2.11. EXAMPLE: BUS RIDERSHIP 19 Aga
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2.12. SIMULATION 21 1 # roll d dice
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2.12. SIMULATION 23 So, in evaluati
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2.12. SIMULATION 25 3 count
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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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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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