A Modern Introduction to Probability and Statistics: Understanding Why and How (Springer Texts in Statistics)

Description

Suitable for self study Use real examples and real data sets that will be familiar to the audience Introduction to the bootstrap is included – this is a modern method missing in many other books Cover ......Page 1 Preface......Page 3 Contents......Page 6 1.1 Biometry: iris recognition......Page 13 1.2 Killer football......Page 15 1.3 Cars and goats: the Monty Hall dilemma......Page 16 1.4 The space shuttle Challenger......Page 17 1.5 Statistics versus intelligence agencies......Page 19 1.6 The speed of light......Page 21 2.1 Sample spaces......Page 25 2.2 Events......Page 26 2.3 Probability......Page 28 2.4 Products of sample spaces......Page 30 2.5 An infinite sample space......Page 31 2.7 Exercises......Page 33 3.1 Conditional probability......Page 37 3.2 The multiplication rule......Page 39 3.3 The law of total probability and Bayes’ rule......Page 42 3.4 Independence......Page 44 3.5 Solutions to the quick exercises......Page 47 3.6 Exercises......Page 49 4.1 Random variables......Page 53 4.2 The probability distribution of a discrete random variable......Page 55 4.3 The Bernoulli and binomial distributions......Page 57 4.4 The geometric distribution......Page 60 4.5 Solutions to the quick exercises......Page 62 4.6 Exercises......Page 63 5.1 Probability density functions......Page 69 5.2 The uniform distribution......Page 72 5.3 The exponential distribution......Page 73 5.4 The Pareto distribution......Page 75 5.5 The normal distribution......Page 76 5.6 Quantiles......Page 77 5.7 Solutions to the quick exercises......Page 79 5.8 Exercises......Page 80 6.1 What is simulation?......Page 83 6.2 Generating realizations of random variables......Page 84 6.3 Comparing two jury rules......Page 87 6.4 The single-server queue......Page 92 6.5 Solutions to the quick exercises......Page 96 6.6 Exercises......Page 97 7.1 Expected values......Page 101 7.2 Three examples......Page 105 7.3 The change-of-variable formula......Page 106 7.4 Variance......Page 108 7.6 Exercises......Page 111 8.1 Transforming discrete random variables......Page 115 8.2 Transforming continuous random variables......Page 116 8.3 Jensen’s inequality......Page 118 8.4 Extremes......Page 120 8.5 Solutions to the quick exercises......Page 122 8.6 Exercises......Page 123 9.1 Joint distributions of discrete random variables......Page 127 9.2 Joint distributions of continuous random variables......Page 130 9.3 More than two random variables......Page 134 9.4 Independent random variables......Page 136 9.5 Propagation of independence......Page 137 9.6 Solutions to the quick exercises......Page 138 9.7 Exercises......Page 139 10.1 Expectation and joint distributions......Page 147 10.2 Covariance......Page 150 10.3 The correlation coefficient......Page 153 10.4 Solutions to the quick exercises......Page 155 10.5 Exercises......Page 156 11.1 Sums of discrete random variables......Page 163 11.2 Sums of continuous random variables......Page 166 11.3 Product and quotient of two random variables......Page 171 11.4 Solutions to the quick exercises......Page 174 11.5 Exercises......Page 175 12.1 Random points......Page 179 12.2 Taking a closer look at random arrivals......Page 180 12.3 The one-dimensional Poisson process......Page 183 12.4 Higher-dimensional Poisson processes......Page 185 12.6 Exercises......Page 188 13.1 Averages vary less......Page 193 13.2 Chebyshev’s inequality......Page 195 13.3 The law of large numbers......Page 197 13.4 Consequences of the law of large numbers......Page 200 13.6 Exercises......Page 203 14.1 Standardizing averages......Page 207 14.2 Applications of the central limit theorem......Page 211 14.3 Solutions to the quick exercises......Page 214 14.4 Exercises......Page 215 15.1 Example: the Old Faithful data......Page 219 15.2 Histograms......Page 221 15.3 Kernel density estimates......Page 224 15.4 The empirical distribution function......Page 231 15.5 Scatterplot......Page 233 15.6 Solutions to the quick exercises......Page 237 15.7 Exercises......Page 238 16.1 The center of a dataset......Page 243 16.2 The amount of variability of a dataset......Page 245 16.3 Empirical quantiles, quartiles, and the IQR......Page 246 16.4 The box-and-whisker plot......Page 248 16.5 Solutions to the quick exercises......Page 250 16.6 Exercises......Page 252 17.1 Random samples and statistical models......Page 257 17.2 Distribution features and sample statistics......Page 260 17.3 Estimating features of the “true” distribution......Page 265 17.4 The linear regression model......Page 268 17.6 Exercises......Page 271 18.1 The bootstrap principle......Page 281 18.2 The empirical bootstrap......Page 284 18.3 The parametric bootstrap......Page 288 18.4 Solutions to the quick exercises......Page 291 18.5 Exercises......Page 292 19.1 Estimators......Page 297 19.2 Investigating the behavior of an estimator......Page 299 19.3 The sampling distribution and unbiasedness......Page 300 19.4 Unbiased estimators for expectation and variance......Page 304 19.6 Exercises......Page 306 20.1 Estimating the number of German tanks......Page 311 20.2 Variance of an estimator......Page 314 20.3 Mean squared error......Page 317 20.5 Exercises......Page 319 21.1 Why a general principle?......Page 325 21.2 The maximum likelihood principle......Page 326 21.3 Likelihood and loglikelihood......Page 328 21.4 Properties of maximum likelihood estimators......Page 333 21.5 Solutions to the quick exercises......Page 334 21.6 Exercises......Page 335 22.1 Least squares estimation and regression......Page 341 22.2 Residuals......Page 344 22.3 Relation with maximum likelihood......Page 347 22.4 Solutions to the quick exercises......Page 348 22.5 Exercises......Page 349 23.1 General principle......Page 353 23.2 Normal data......Page 357 23.3 Bootstrap confidence intervals......Page 362 23.4 Large samples......Page 365 23.5 Solutions to the quick exercises......Page 367 23.6 Exercises......Page 368 24.1 The probability of success......Page 373 24.2 Is there a general method?......Page 376 24.3 One-sided confidence intervals......Page 378 24.4 Determining the sample size......Page 379 24.5 Solutions to the quick exercises......Page 380 24.6 Exercises......Page 381 25.1 Null hypothesis and test statistic......Page 385 25.2 Tail probabilities......Page 388 25.3 Type I and type II errors......Page 389 25.4 Solutions to the quick exercises......Page 391 25.5 Exercises......Page 392 26.1 Significance level......Page 395 26.2 Critical region and critical values......Page 398 26.3 Type II error......Page 402 26.4 Relation with confidence intervals......Page 404 26.5 Solutions to the quick exercises......Page 405 26.6 Exercises......Page 406 27.1 Monitoring the production of ball bearings......Page 411 27.2 The one-sample t-test......Page 413 27.3 The t-test in a regression setting......Page 417 27.4 Solutions to the quick exercises......Page 421 27.5 Exercises......Page 422 28.1 Is dry drilling faster than wet drilling?......Page 427 28.2 Two samples with equal variances......Page 428 28.3 Two samples with unequal variances......Page 431 28.4 Large samples......Page 434 28.6 Exercises......Page 436 A Summary of distributions......Page 441 B Tables of the normal and t-distributions......Page 443 C Answers to selected exercises......Page 447 D Full solutions to selected exercises......Page 457 Index......Page 0