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114 CHAPTER 5. CONTINUOUS PROBABILITY MODELS > 1 − pchisq ( 0 . 3 6 / 0 . 0 4 , 2 ) [ 1 ] 0.01110900 5.5.3.3 Importance in Modeling This distribution is used widely in statistical applications. As will be seen in our chapters on statistics, many statistical methods involve a sum of squared normal random variables. 7 5.5.4 The Exponential Family of Distributions Please note: We have been talking here of parametric families of distributions, and in this section will introduce one of the most famous, the family of exponential distributions. This should not be confused, though, with the term exponential family that arises in mathematical statistics, which includes exponential distributions but is much broader. 5.5.4.1 Density and Properties The densities in this family have the form This is a one-parameter family of distributions. fW (t) = λe −λt , 0 < t < ∞ (5.66) After integration, one finds that E(W ) = 1 1 λ and V ar(W ) = λ2 . You might wonder why it is customary to index the family via λ rather than 1/λ (see (5.66)), since the latter is the mean. But this is actually quite natural, for the reason cited in the following subsection. 5.5.4.2 R Functions Relevant functions for a uniformly distributed random variable X with parameter λ are • pexp(q,lambda), to find P (X ≤ q) • qexp(q,lambda), to find c such that P (X ≤ c) = q • rexp(n,lambda), to generate n independent values of X 7 The motivation for the term degrees of freedom will be explained in those chapters too.
5.5. FAMOUS PARAMETRIC FAMILIES OF CONTINUOUS DISTRIBUTIONS 115 5.5.4.3 Example: Refunds on Failed Components Suppose a manufacturer of some electronic component finds that its lifetime L is exponentially distributed with mean 10000 hours. They give a refund if the item fails before 500 hours. Let M be the number of items they have sold, up to and including the one on which they make the first refund. Let’s find EM and V ar(M). First, notice that M has a geometric distribution! It is the number of independent trials until the first success, where a “trial” is one component, “success” (no value judgment, remember) is giving a refund, and the success probability is P (L < 500) = Then plug p = 0.05 into (3.87) and (3.88). 500 5.5.4.4 Example: Overtime Parking Fees 0 0.0001e −0.0001t dt = 0.05 (5.67) A certain public parking garage charges parking fees of $1.50 for the first hour, and $1 per hour after that. Suppose parking times T are exponentially distributed with mean 1.5 hours. Let W denote the total fee paid. Let’s find E(W) and Var(W). The key point is that W is a function of T: W = 1.5T, if T ≤ 1 1.5 + 1 · (T − 1) = T + 0.5, if T > 1 (5.68) That’s good, because we know how to find the expected value of a function of a continuous random variable, from (5.24). Defining g() as in (5.68) above, we have ∞ EW = 0 g(t) 1 1 e− 1.5 1.5 t dt = The integration is left to the reader. 1 0 1.5t 1 1 e− 1.5 1.5 t dt + ∞ 1 (t + 0.5) 1 1 e− 1.5 1.5 t dt (5.69) Now, what about Var(W)? As is often the case, it’s easier to use (3.30), so we need to find E(W 2 ). The above integration becomes E(W 2 ∞ ) = 0 g 2 (t) 1 1 e− 1.5 1.5 t dt = 1 0 1.5 2 t 1 1 e− 1.5 1.5 t dt + ∞ 2 1 (t + 0.5) 1.5 1 e− 1 1.5 t dt (5.70)
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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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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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