Preface |
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xiii | |
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1 Probability and counting |
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1 | (44) |
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1.1 Why study probability? |
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1 | (2) |
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1.2 Sample spaces and Pebble World |
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3 | (3) |
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1.3 Naive definition of probability |
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6 | (2) |
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8 | (12) |
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20 | (1) |
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1.6 Non-naive definition of probability |
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21 | (5) |
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26 | (3) |
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29 | (4) |
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33 | (12) |
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2 Conditional probability |
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45 | (58) |
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2.1 The importance of thinking conditionally |
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45 | (1) |
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2.2 Definition and intuition |
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46 | (6) |
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2.3 Bayes' rule and the law of total probability |
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52 | (7) |
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2.4 Conditional probabilities are probabilities |
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59 | (4) |
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2.5 Independence of events |
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63 | (4) |
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2.6 Coherency of Bayes' rule |
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67 | (1) |
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2.7 Conditioning as a problem-solving tool |
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68 | (6) |
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2.8 Pitfalls and paradoxes |
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74 | (5) |
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79 | (1) |
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80 | (3) |
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83 | (20) |
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3 Random variables and their distributions |
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103 | (46) |
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103 | (3) |
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3.2 Distributions and probability mass functions |
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106 | (6) |
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3.3 Bernoulli and Binomial |
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112 | (3) |
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115 | (3) |
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118 | (2) |
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3.6 Cumulative distribution functions |
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120 | (3) |
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3.7 Functions of random variables |
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123 | (6) |
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3.8 Independence of r.v.s |
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129 | (4) |
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3.9 Connections between Binomial and Hypergeometric |
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133 | (3) |
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136 | (2) |
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138 | (2) |
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140 | (9) |
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149 | (64) |
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4.1 Definition of expectation |
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149 | (3) |
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4.2 Linearity of expectation |
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152 | (5) |
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4.3 Geometric and Negative Binomial |
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157 | (7) |
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4.4 Indicator r.v.s and the fundamental bridge |
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164 | (6) |
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4.5 Law of the unconscious statistician (LOTUS) |
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170 | (1) |
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171 | (3) |
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174 | (7) |
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4.8 Connections between Poisson and Binomial |
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181 | (3) |
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4.9 *Using probability and expectation to prove existence |
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184 | (5) |
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189 | (3) |
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192 | (2) |
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194 | (19) |
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5 Continuous random variables |
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213 | (54) |
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5.1 Probability density functions |
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213 | (7) |
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220 | (4) |
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5.3 Universality of the Uniform |
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224 | (7) |
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231 | (7) |
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238 | (6) |
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244 | (4) |
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5.7 Symmetry of i.i.d. continuous r.v.s |
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248 | (2) |
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250 | (3) |
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253 | (2) |
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255 | (12) |
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267 | (36) |
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6.1 Summaries of a distribution |
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267 | (5) |
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272 | (4) |
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276 | (3) |
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6.4 Moment generating functions |
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279 | (4) |
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6.5 Generating moments with MGFs |
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283 | (3) |
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6.6 Sums of independent r.v.s via MGFs |
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286 | (1) |
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6.7 *Probability generating functions |
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287 | (5) |
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292 | (1) |
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293 | (5) |
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298 | (5) |
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303 | (64) |
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7.1 Joint, marginal, and conditional |
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304 | (20) |
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324 | (2) |
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7.3 Covariance and correlation |
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326 | (6) |
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332 | (11) |
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7.5 Multivariate Normal . 33? |
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343 | (3) |
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346 | (2) |
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348 | (19) |
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367 | (48) |
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369 | (6) |
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375 | (4) |
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379 | (8) |
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387 | (9) |
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8.5 Beta-Gamma connections |
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396 | (2) |
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398 | (4) |
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402 | (2) |
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404 | (3) |
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407 | (8) |
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9 Conditional expectation |
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415 | (42) |
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9.1 Conditional expectation given an event |
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415 | (9) |
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9.2 Conditional expectation given an r.v. |
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424 | (2) |
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9.3 Properties of conditional expectation |
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426 | (5) |
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9.4 *Geometric interpretation of conditional expectation |
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431 | (1) |
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432 | (4) |
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9.6 Adam and Eve examples |
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436 | (3) |
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439 | (2) |
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441 | (2) |
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443 | (14) |
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10 Inequalities and limit theorems |
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457 | (40) |
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458 | (9) |
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10.2 Law of large numbers |
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467 | (4) |
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10.3 Central limit theorem |
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471 | (6) |
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10.4 Chi-Square and Student |
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477 | (3) |
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480 | (3) |
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483 | (3) |
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486 | (11) |
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497 | (38) |
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11.1 Markov property and transit ion matrix |
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497 | (5) |
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11.2 Classification of states |
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502 | (4) |
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11.3 Stationary distribution |
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506 | (7) |
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513 | (7) |
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520 | (1) |
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521 | (3) |
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524 | (11) |
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12 Markov chain Monte Carlo |
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535 | (24) |
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536 | (12) |
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548 | (6) |
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554 | (1) |
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555 | (2) |
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557 | (2) |
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559 | (22) |
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13.1 Poisson processes in one dimension |
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559 | (2) |
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13.2 Conditioning, superposition, and thinning |
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561 | (12) |
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13.3 Poisson processes in multiple dimensions |
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573 | (2) |
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575 | (1) |
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575 | (2) |
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577 | (4) |
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581 | (20) |
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581 | (4) |
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585 | (5) |
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590 | (2) |
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592 | (1) |
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A.5 Differential equations |
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593 | (1) |
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594 | (1) |
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594 | (2) |
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596 | (3) |
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599 | (1) |
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A.10 Common sense and checking answers |
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599 | (2) |
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601 | (6) |
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601 | (1) |
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602 | (1) |
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602 | (1) |
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B.4 Sampling and simulation |
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603 | (1) |
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603 | (1) |
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603 | (1) |
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604 | (1) |
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604 | (1) |
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605 | (2) |
References |
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607 | (2) |
Index |
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609 | |