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.
Author(s): F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester
Series: Springer Texts in Statistics
Publisher: Springer
Year: 2005
Language: English
Pages: 483
A Modern Introduction to Probability and Statistics......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 24
2.2 Events......Page 25
2.3 Probability......Page 27
2.4 Products of sample spaces......Page 29
2.5 An infinite sample space......Page 30
2.7 Exercises......Page 32
3.1 Conditional probability......Page 36
3.2 The multiplication rule......Page 38
3.3 The law of total probability and Bayes’ rule......Page 41
3.4 Independence......Page 43
3.5 Solutions to the quick exercises......Page 46
3.6 Exercises......Page 48
4.1 Random variables......Page 52
4.2 The probability distribution of a discrete random variable......Page 54
4.3 The Bernoulli and binomial distributions......Page 56
4.4 The geometric distribution......Page 59
4.5 Solutions to the quick exercises......Page 61
4.6 Exercises......Page 62
5.1 Probability density functions......Page 67
5.2 The uniform distribution......Page 70
5.3 The exponential distribution......Page 71
5.4 The Pareto distribution......Page 73
5.5 The normal distribution......Page 74
5.6 Quantiles......Page 75
5.7 Solutions to the quick exercises......Page 77
5.8 Exercises......Page 78
6.1 What is simulation?......Page 81
6.2 Generating realizations of random variables......Page 82
6.3 Comparing two jury rules......Page 85
6.4 The single-server queue......Page 90
6.5 Solutions to the quick exercises......Page 94
6.6 Exercises......Page 95
7.1 Expected values......Page 98
7.2 Three examples......Page 102
7.3 The change-of-variable formula......Page 103
7.4 Variance......Page 105
7.6 Exercises......Page 108
8.1 Transforming discrete random variables......Page 112
8.2 Transforming continuous random variables......Page 113
8.3 Jensen’s inequality......Page 115
8.5 Solutions to the quick exercises......Page 119
8.6 Exercises......Page 120
9.1 Joint distributions of discrete random variables......Page 124
9.2 Joint distributions of continuous random variables......Page 127
9.3 More than two random variables......Page 131
9.4 Independent random variables......Page 133
9.5 Propagation of independence......Page 134
9.6 Solutions to the quick exercises......Page 135
9.7 Exercises......Page 136
10.1 Expectation and joint distributions......Page 144
10.2 Covariance......Page 147
10.3 The correlation coefficient......Page 150
10.4 Solutions to the quick exercises......Page 152
10.5 Exercises......Page 153
11.1 Sums of discrete random variables......Page 160
11.2 Sums of continuous random variables......Page 163
11.3 Product and quotient of two random variables......Page 168
11.4 Solutions to the quick exercises......Page 171
11.5 Exercises......Page 172
12.1 Random points......Page 176
12.2 Taking a closer look at random arrivals......Page 177
12.3 The one-dimensional Poisson process......Page 180
12.4 Higher-dimensional Poisson processes......Page 182
12.6 Exercises......Page 185
13.1 Averages vary less......Page 189
13.2 Chebyshev’s inequality......Page 191
13.3 The law of large numbers......Page 193
13.4 Consequences of the law of large numbers......Page 196
13.6 Exercises......Page 199
14.1 Standardizing averages......Page 203
14.2 Applications of the central limit theorem......Page 207
14.3 Solutions to the quick exercises......Page 210
14.4 Exercises......Page 211
15.1 Example: the Old Faithful data......Page 214
15.2 Histograms......Page 216
15.3 Kernel density estimates......Page 219
15.4 The empirical distribution function......Page 226
15.5 Scatterplot......Page 228
15.6 Solutions to the quick exercises......Page 232
15.7 Exercises......Page 233
16.1 The center of a dataset......Page 238
16.2 The amount of variability of a dataset......Page 240
16.3 Empirical quantiles, quartiles, and the IQR......Page 241
16.4 The box-and-whisker plot......Page 243
16.5 Solutions to the quick exercises......Page 245
16.6 Exercises......Page 247
17.1 Random samples and statistical models......Page 251
17.2 Distribution features and sample statistics......Page 254
17.3 Estimating features of the “true” distribution......Page 259
17.4 The linear regression model......Page 262
17.6 Exercises......Page 265
18.1 The bootstrap principle......Page 275
18.2 The empirical bootstrap......Page 278
18.3 The parametric bootstrap......Page 282
18.4 Solutions to the quick exercises......Page 285
18.5 Exercises......Page 286
19.1 Estimators......Page 291
19.2 Investigating the behavior of an estimator......Page 293
19.3 The sampling distribution and unbiasedness......Page 294
19.4 Unbiased estimators for expectation and variance......Page 298
19.6 Exercises......Page 300
20.1 Estimating the number of German tanks......Page 304
20.2 Variance of an estimator......Page 307
20.3 Mean squared error......Page 310
20.5 Exercises......Page 312
21.1 Why a general principle?......Page 317
21.2 The maximum likelihood principle......Page 318
21.3 Likelihood and loglikelihood......Page 320
21.4 Properties of maximum likelihood estimators......Page 325
21.5 Solutions to the quick exercises......Page 326
21.6 Exercises......Page 327
22.1 Least squares estimation and regression......Page 332
22.2 Residuals......Page 335
22.3 Relation with maximum likelihood......Page 338
22.4 Solutions to the quick exercises......Page 339
22.5 Exercises......Page 340
23.1 General principle......Page 344
23.2 Normal data......Page 348
23.3 Bootstrap confidence intervals......Page 353
23.4 Large samples......Page 356
23.5 Solutions to the quick exercises......Page 358
23.6 Exercises......Page 359
24.1 The probability of success......Page 364
24.2 Is there a general method?......Page 367
24.3 One-sided confidence intervals......Page 369
24.4 Determining the sample size......Page 370
24.5 Solutions to the quick exercises......Page 371
24.6 Exercises......Page 372
25.1 Null hypothesis and test statistic......Page 375
25.2 Tail probabilities......Page 378
25.3 Type I and type II errors......Page 379
25.4 Solutions to the quick exercises......Page 381
25.5 Exercises......Page 382
26.1 Significance level......Page 385
26.2 Critical region and critical values......Page 388
26.3 Type II error......Page 392
26.4 Relation with confidence intervals......Page 394
26.5 Solutions to the quick exercises......Page 395
26.6 Exercises......Page 396
27.1 Monitoring the production of ball bearings......Page 400
27.2 The one-sample t-test......Page 402
27.3 The t-test in a regression setting......Page 406
27.4 Solutions to the quick exercises......Page 410
27.5 Exercises......Page 411
28.1 Is dry drilling faster than wet drilling?......Page 415
28.2 Two samples with equal variances......Page 416
28.3 Two samples with unequal variances......Page 419
28.4 Large samples......Page 422
28.6 Exercises......Page 424
Continuous distributions......Page 429
B Tables of the normal and t-distributions......Page 431
Table B.1. Right tail probabilities......Page 432
Table B.2. Right critical values......Page 433
C Answers to selected exercises......Page 434
D Full solutions to selected exercises......Page 443
References......Page 472
List of symbols......Page 474
Index......Page 476
