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): Henning Mortveit, Christian Reidys
Series: Texts in Statistics
Publisher: Springer
Year: 2005
Language: English
Pages: 485
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
