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436 CHAPTER 21. INTRODUCTION TO QUEUING MODELS The system always reserves g channels for handoff calls. When a request for a new call (i.e. a non-handoff call) arrives, the system looks at Xt, the current number of calls in the cell. If that number is less than n-g, so that there are more than g idle channels available, the new call is accepted; otherwise it is rejected. We assume that new calls originate from within the cells according to a Poisson process with rate λ1, while handoff calls drift in from neighboring cells at rate λ2. Meanwhile, call durations are exponential with rate µ1, while the time that a call remains within the cell is exponential with rate µ2. 21.4.1.1 Stationary Distribution We again have a birth/death process, though a bit more complicated than our earlier ones. Let λ = λ1 + λ2 and µ = µ1 + µ2. Then here is a sample balance equation, focused on transitions into (left-hand side in the equation) and out of (right-hand side) state 1: π0λ + π22µ = π1(λ + µ) (21.43) Here’s why: How can we enter state 1? Well, we could do so from state 0, where there are no calls; this occurs if we get a new call (rate λ1) or a handoff call (rate λ2. In state 2, we enter state 1 if one of the two calls ends (rate µ1) or one of the two calls leaves the cell (rate µ2). The same kind of reasoning shows that we leave state 1 at rate λ + µ. As another example, here is the equation for state n-g: πn−g[λ2 + (n − g)µ] = πn−g+1 · (n − g + 1)µ + πn−g−1λ (21.44) Note the term λ2 in (21.44), rather than λ as in (21.43). Using our birth/death formula for the πi, we find that where A = λ/µ, A1 = λ2/µ and πk = ⎡ π0 = ⎣ π0 Ak k! , k ≤ n-g π0 An−g k! Ak−(n−g) 1 , k ≥ n-g n−g−1 k=0 A k k! + n k=n−g A n−g k! Ak−(n−g) 1 ⎤ ⎦ −1 (21.45) (21.46)
21.5. NONEXPONENTIAL SERVICE TIMES 437 21.4.1.2 Going Beyond Finding the π One can calculate a number of interesting quantities from the πi: • The probability of a handoff call being rejected is πn. • The probability of a new call being dropped is n k=n−g πk (21.47) • Since the per-channel utilization in state i is i/n, the overall long-run per-channel utilization is n i=0 πi i n (21.48) • The long-run proportion of accepted calls which are handoff calls is the rate at which handoff calls are accepted, divided by the rate at which calls are accepted: λ1 n−1 λ2 i=0 πi n−g−1 i=0 πi + λ2 21.5 Nonexponential Service Times n−1 i=0 πi (21.49) The Markov property is of course crucial to the analyses we made above. Thus dropping the exponential assumption presents a major analytical challenge. One queuing model which has been found tractable is M/G/1: Exponential interarrival times, general service times, one server. In fact, the mean queue length and related quantities can be obtained fairly easily, as follows. Consider the residence time R for a tagged job. R is the time that our tagged job must first wait for completion of service of all jobs, if any, which are ahead of it—queued or now in service—plus the tagged job’s own service time. Let T1, T2, ... be i.i.d. with the distribution of a generic service time random variable S. T1 represents the service time of the tagged job itself. T2, ..., TN represent the service times of the queued jobs, if any. Let N be the number of jobs in the system, either being served or queued; B be either 1 or 0, depending on whether the system is busy (i.e. N > 0) or not; and S1,resid be the remaining service
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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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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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Page 473 and 474: Appendix A Review of Matrix Algebra
Page 475 and 476: A.2. MATRIX TRANSPOSE 451 • Matri
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Page 487 and 488: B.6. COMPLEX NUMBERS 463 B.6 Comple
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