Preface |
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xi | |
Acknowledgments |
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xv | |
About the Companion Website |
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xvii | |
Introduction |
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xix | |
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1 | (44) |
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1.1 Random Experiment, Sample Space, Event |
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1 | (2) |
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1.2 What Is a Probability? |
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3 | (1) |
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4 | (3) |
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1.4 Properties of Probabilities |
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7 | (4) |
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1.5 Equally Likely Outcomes |
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11 | (1) |
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12 | (4) |
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13 | (3) |
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16 | (10) |
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1.7.1 Combinations and Binomial Coefficients |
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17 | (9) |
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1.8 Problem-Solving Strategies: Complements and Inclusion-Exclusion |
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26 | (3) |
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1.9 A First Look at Simulation |
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29 | (5) |
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34 | (11) |
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36 | (9) |
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2 Conditional Probability and Independence |
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45 | (48) |
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2.1 Conditional Probability |
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45 | (5) |
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2.2 New Information Changes the Sample Space |
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50 | (1) |
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51 | (9) |
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56 | (4) |
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2.4 Conditioning and the Law of Total Probability |
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60 | (7) |
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2.5 Bayes Formula and Inverting a Conditional Probability |
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67 | (5) |
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2.6 Independence and Dependence |
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72 | (8) |
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80 | (2) |
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82 | (11) |
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83 | (10) |
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3 Introduction to Discrete Random Variables |
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93 | (32) |
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93 | (4) |
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3.2 Independent Random Variables |
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97 | (2) |
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99 | (2) |
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3.4 Binomial Distribution |
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101 | (7) |
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108 | (8) |
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3.5.1 Poisson Approximation of Binomial Distribution |
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113 | (2) |
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3.5.2 Poisson as Limit of Binomial Probabilities |
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115 | (1) |
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116 | (9) |
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118 | (7) |
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4 Expectation and More with Discrete Random Variables |
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125 | (60) |
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127 | (3) |
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4.2 Functions of Random Variables |
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130 | (4) |
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134 | (5) |
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4.4 Independent Random Variables |
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139 | (5) |
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4.4.1 Sums of Independent Random Variables |
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142 | (2) |
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4.5 Linearity of Expectation |
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144 | (5) |
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4.6 Variance and Standard Deviation |
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149 | (9) |
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4.7 Covariance and Correlation |
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158 | (7) |
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4.8 Conditional Distribution |
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165 | (6) |
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4.8.1 Introduction to Conditional Expectation |
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168 | (3) |
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4.9 Properties of Covariance and Correlation |
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171 | (2) |
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4.10 Expectation of a Function of a Random Variable |
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173 | (1) |
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174 | (11) |
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176 | (9) |
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5 More Discrete Distributions and Their Relationships |
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185 | (42) |
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5.1 Geometric Distribution |
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185 | (8) |
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188 | (1) |
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5.1.2 Coupon Collecting and Tiger Counting |
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189 | (4) |
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5.2 Moment-Generating Functions |
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193 | (3) |
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5.3 Negative Binomial--Up from the Geometric |
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196 | (6) |
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5.4 Hypergeometric--Sampling Without Replacement |
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202 | (5) |
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5.5 From Binomial to Multinomial |
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207 | (6) |
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213 | (3) |
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216 | (11) |
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218 | (9) |
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227 | (46) |
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6.1 Probability Density Function |
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229 | (4) |
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6.2 Cumulative Distribution Function |
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233 | (4) |
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6.3 Expectation and Variance |
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237 | (2) |
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239 | (3) |
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6.5 Exponential Distribution |
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242 | (5) |
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243 | (4) |
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247 | (9) |
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256 | (6) |
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6.7.1 Accept-Reject Method |
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258 | (4) |
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6.8 Covariance, Correlation |
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262 | (2) |
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264 | (9) |
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266 | (7) |
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7 Continuous Distributions |
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273 | (46) |
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273 | (15) |
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7.1.1 Standard Normal Distribution |
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276 | (2) |
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7.1.2 Normal Approximation of Binomial Distribution |
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278 | (4) |
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282 | (3) |
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7.1.4 Sums of Independent Normals |
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285 | (3) |
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288 | (6) |
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7.2.1 Probability as a Technique of Integration |
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292 | (2) |
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294 | (8) |
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302 | (3) |
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305 | (3) |
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308 | (11) |
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311 | (8) |
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8 Densities of Functions of Random Variables |
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319 | (38) |
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320 | (10) |
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8.1.1 Simulating a Continuous Random Variable |
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326 | (3) |
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8.1.2 Method of Transformations |
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329 | (1) |
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8.2 Maximums, Minimums, and Order Statistics |
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330 | (5) |
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335 | (3) |
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8.4 Geometric Probability |
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338 | (6) |
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8.5 Transformations of Two Random Variables |
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344 | (4) |
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348 | (9) |
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349 | (8) |
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9 Conditional Distribution, Expectation, and Variance |
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357 | (50) |
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357 | (1) |
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9.1 Conditional Distributions |
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358 | (6) |
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9.2 Discrete and Continuous: Mixing it Up |
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364 | (5) |
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9.3 Conditional Expectation |
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369 | (9) |
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9.3.1 From Function to Random Variable |
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371 | (7) |
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9.3.2 Random Sum of Random Variables |
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378 | (1) |
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9.4 Computing Probabilities by Conditioning |
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378 | (4) |
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382 | (5) |
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9.6 Bivariate Normal Distribution |
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387 | (9) |
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396 | (11) |
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398 | (9) |
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407 | (40) |
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10.1 Weak Law of Large Numbers |
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409 | (6) |
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10.1.1 Markov and Chebyshev Inequalities |
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411 | (4) |
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10.2 Strong Law of Large Numbers |
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415 | (6) |
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421 | (3) |
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10.4 Monte Carlo Integration |
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424 | (4) |
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10.5 Central Limit Theorem |
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428 | (9) |
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10.5.1 Central Limit Theorem and Monte Carlo |
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436 | (1) |
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10.6 A Proof of the Central Limit Theorem |
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437 | (2) |
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439 | (8) |
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440 | (7) |
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11 Beyond Random Walks and Markov Chains |
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447 | (34) |
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11.1 Random Walks on Graphs |
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447 | (8) |
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11.1.1 Long-Term Behavior |
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451 | (4) |
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11.2 Random Walks on Weighted Graphs and Markov Chains |
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455 | (7) |
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11.2.1 Stationary Distribution |
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458 | (4) |
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11.3 From Markov Chain to Markov Chain Monte Carlo |
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462 | (12) |
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474 | (7) |
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476 | (5) |
Appendix A Probability Distributions in R |
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481 | (2) |
Appendix B Summary of Probability Distributions |
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483 | (4) |
Appendix C Mathematical Reminders |
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487 | (2) |
Appendix D Working with Joint Distributions |
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489 | (8) |
Solutions to Exercises |
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497 | (14) |
References |
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511 | (4) |
Index |
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515 | |