A Modern Introduction to Probability and Statistics: Understanding Why and How

Description

Probability and Statistics are studied by most science students. Many current texts in the area are just cookbooks and, as a result, students do not know why they perform the methods they are taught, or why the methods work. The strength of this book is that it readdresses these shortcomings; by using examples, often from real-life and using real data, the authors show how the fundamentals of probabilistic and statistical theories arise intuitively. A Modern Introduction to Probability and Statistics has numerous quick exercises to give direct feedback to students. In addition there are over 350 exercises, half of which have answers, of which half have full solutions. A website gives access to the data files used in the text, and, for instructors, the remaining solutions. The only pre-requisite is a first course in calculus; the text covers standard statistics and probability material, and develops beyond traditional parametric models to the Poisson process, and on to modern methods such as the bootstrap. Cover......Page 1 A Modern Introduction to Probability and Statistics......Page 3 Preface......Page 5 Contents......Page 8 1.1 Biometry: iris recognition......Page 15 1.2 Killer football......Page 17 1.3 Cars and goats: the Monty Hall dilemma......Page 18 1.4 The space shuttle Challenger......Page 19 1.5 Statistics versus intelligence agencies......Page 21 1.6 The speed of light......Page 23 2.1 Sample spaces......Page 26 2.2 Events......Page 27 2.3 Probability......Page 29 2.4 Products of sample spaces......Page 31 2.5 An infinite sample space......Page 32 2.7 Exercises......Page 34 3.1 Conditional probability......Page 38 3.2 The multiplication rule......Page 40 3.3 The law of total probability and Bayes’ rule......Page 43 3.4 Independence......Page 45 3.5 Solutions to the quick exercises......Page 48 3.6 Exercises......Page 50 4.1 Random variables......Page 54 4.2 The probability distribution of a discrete random variable......Page 56 4.3 The Bernoulli and binomial distributions......Page 58 4.4 The geometric distribution......Page 61 4.5 Solutions to the quick exercises......Page 63 4.6 Exercises......Page 64 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 100 7.2 Three examples......Page 104 7.3 The change-of-variable formula......Page 105 7.4 Variance......Page 107 7.6 Exercises......Page 110 8.1 Transforming discrete random variables......Page 114 8.2 Transforming continuous random variables......Page 115 8.3 Jensen’s inequality......Page 117 8.5 Solutions to the quick exercises......Page 121 8.6 Exercises......Page 122 9.1 Joint distributions of discrete random variables......Page 126 9.2 Joint distributions of continuous random variables......Page 129 9.3 More than two random variables......Page 133 9.4 Independent random variables......Page 135 9.5 Propagation of independence......Page 136 9.6 Solutions to the quick exercises......Page 137 9.7 Exercises......Page 138 10.1 Expectation and joint distributions......Page 146 10.2 Covariance......Page 149 10.3 The correlation coefficient......Page 152 10.4 Solutions to the quick exercises......Page 154 10.5 Exercises......Page 155 11.1 Sums of discrete random variables......Page 162 11.2 Sums of continuous random variables......Page 165 11.3 Product and quotient of two random variables......Page 170 11.4 Solutions to the quick exercises......Page 173 11.5 Exercises......Page 174 12.1 Random points......Page 178 12.2 Taking a closer look at random arrivals......Page 179 12.3 The one-dimensional Poisson process......Page 182 12.4 Higher-dimensional Poisson processes......Page 184 12.6 Exercises......Page 187 13.1 Averages vary less......Page 191 13.2 Chebyshev’s inequality......Page 193 13.3 The law of large numbers......Page 195 13.4 Consequences of the law of large numbers......Page 198 13.6 Exercises......Page 201 14.1 Standardizing averages......Page 205 14.2 Applications of the central limit theorem......Page 209 14.3 Solutions to the quick exercises......Page 212 14.4 Exercises......Page 213 15.1 Example: the Old Faithful data......Page 216 15.2 Histograms......Page 218 15.3 Kernel density estimates......Page 221 15.4 The empirical distribution function......Page 228 15.5 Scatterplot......Page 230 15.6 Solutions to the quick exercises......Page 234 15.7 Exercises......Page 235 16.1 The center of a dataset......Page 240 16.2 The amount of variability of a dataset......Page 242 16.3 Empirical quantiles, quartiles, and the IQR......Page 243 16.4 The box-and-whisker plot......Page 245 16.5 Solutions to the quick exercises......Page 247 16.6 Exercises......Page 249 17.1 Random samples and statistical models......Page 253 17.2 Distribution features and sample statistics......Page 256 17.3 Estimating features of the “true” distribution......Page 261 17.4 The linear regression model......Page 264 17.6 Exercises......Page 267 18.1 The bootstrap principle......Page 277 18.2 The empirical bootstrap......Page 280 18.3 The parametric bootstrap......Page 284 18.4 Solutions to the quick exercises......Page 287 18.5 Exercises......Page 288 19.1 Estimators......Page 293 19.2 Investigating the behavior of an estimator......Page 295 19.3 The sampling distribution and unbiasedness......Page 296 19.4 Unbiased estimators for expectation and variance......Page 300 19.6 Exercises......Page 302 20.1 Estimating the number of German tanks......Page 306 20.2 Variance of an estimator......Page 309 20.3 Mean squared error......Page 312 20.5 Exercises......Page 314 21.1 Why a general principle?......Page 319 21.2 The maximum likelihood principle......Page 320 21.3 Likelihood and loglikelihood......Page 322 21.4 Properties of maximum likelihood estimators......Page 327 21.5 Solutions to the quick exercises......Page 328 21.6 Exercises......Page 329 22.1 Least squares estimation and regression......Page 334 22.2 Residuals......Page 337 22.3 Relation with maximum likelihood......Page 340 22.4 Solutions to the quick exercises......Page 341 22.5 Exercises......Page 342 23.1 General principle......Page 346 23.2 Normal data......Page 350 23.3 Bootstrap confidence intervals......Page 355 23.4 Large samples......Page 358 23.5 Solutions to the quick exercises......Page 360 23.6 Exercises......Page 361 24.1 The probability of success......Page 366 24.2 Is there a general method?......Page 369 24.3 One-sided confidence intervals......Page 371 24.4 Determining the sample size......Page 372 24.5 Solutions to the quick exercises......Page 373 24.6 Exercises......Page 374 25.1 Null hypothesis and test statistic......Page 377 25.2 Tail probabilities......Page 380 25.3 Type I and type II errors......Page 381 25.4 Solutions to the quick exercises......Page 383 25.5 Exercises......Page 384 26.1 Significance level......Page 387 26.2 Critical region and critical values......Page 390 26.3 Type II error......Page 394 26.4 Relation with confidence intervals......Page 396 26.5 Solutions to the quick exercises......Page 397 26.6 Exercises......Page 398 27.1 Monitoring the production of ball bearings......Page 402 27.2 The one-sample t-test......Page 404 27.3 The t-test in a regression setting......Page 408 27.4 Solutions to the quick exercises......Page 412 27.5 Exercises......Page 413 28.1 Is dry drilling faster than wet drilling?......Page 417 28.2 Two samples with equal variances......Page 418 28.3 Two samples with unequal variances......Page 421 28.4 Large samples......Page 424 28.6 Exercises......Page 426 Continuous distributions......Page 431 B Tables of the normal and t-distributions......Page 433 Table B.1. Right tail probabilities......Page 434 Table B.2. Right critical values......Page 435 C Answers to selected exercises......Page 436 D Full solutions to selected exercises......Page 445 References......Page 474 List of symbols......Page 476 Index......Page 478