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190 CHAPTER 9. INTRODUCTION TO CONTINUOUS-TIME MARKOV CHAINS Of course, we also have the constraint π2 + π1 + π0 = 1. The solution turns out to be π = (0.072, 0.362, 0.566) (9.26) Thus for example, during 7.2% of the time, there will be no machine available at all. Several variations of this problem could be analyzed. We could compare the two-machine system with a one-machine version. It turns out that the proportion of down time (i.e. time when no machine is available) increases to 28.6%. Or we could analyze the case in which only one repair person is employed by this factory, so that only one machine can be repaired at a time, compared to the situation above, in which we (tacitly) assumed that if both machines are down, they can be repaired in parallel. We leave these variations as exercises for the reader. 9.3.4 Example: Migration in a Social Network The following is a simplified version of research in online social networks. There is a town with two social groups, with each person being in exactly one group. People arrive from outside town, with exponentially distributed interarrival times at rate α, and join one of the groups with probability 0.5 each. Each person will occasionally switch groups, with one possible “switch” being to leave town entirely. A person’s time before switching is exponentially distributed with rate σ; the switch will either be to the other group or to the outside world, with probabilities q and 1-q, respectively. Let the state of the system be (i,j), where i and j are the number of current members in groups 1 and 2, respectively. Let’s find a typical balance equation, say for the state (8,8): π (8,8)(α + 16 · σ) = (π (9,8) + π (8,9)) · 9σ(1 − q) + (π (9,7) + π (7,9)) · 9σq + (π (8,7) + π (7,8)) · 0.5α (9.27) The reasoning is straightforward. How can we move out of state (8,8)? Well, there could be an arrival (rate α), or any one of the 16 people could switch groups (rate 16σ), etc. Now, in a “going beyond finding the π” vein, let’s find the long-run fraction of transfers into group 1 that come from group 2, as opposed to from the outside. The rate of transitions into that group from outside is 0.5α. When the system is in state (i,j), the rate of transitions into group 1 from group 2 is jσq, so the overall rate is i,j π (i,j)jσq. Thus the fraction of new members coming in to group 1 from transfers is i,j π (i,j)jσq α + i,j π (i,j)jσq (9.28)
9.3. HOLDING-TIME DISTRIBUTION 191 The above reasoning is very common, quite applicable in many situations. By the way, note that i,j π (i,j)jσq = σqEN, where N is the number of members of group 1.
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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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Page 249 and 250: 11.5. ONE-SIDED HA 225 By checking
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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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