A Modern Introduction To Probability And Statistics: Understanding Why and How

by Dekking, F. M.; Kraaikamp, C.; Lopuhaa, H. P.; Meester, L. E.; Dekking, Michel

• ISBN13: 9781852338961
• ISBN10: 1852338962
• eBook ISBN(s): 9781846281686
• Format: Hardcover
• Copyright: 2005-05-23
• Publisher: Springer Verlag
• More Book Details

Summary

Probability and Statistics are studied by most science students, usually as a second- or third-year course. 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 can show how the fundamentals of probabilistic and statistical theories arise intuitively. It provides a tried and tested, self-contained course, that can also be used for self-study. A Modern Introduction to Probability and Statistics has numerous quick exercises to give direct feedback to the students. In addition the book contains over 350 exercises, half of which have answers, of which half have full solutions. A website at www.springeronline.com/1-85233-896-2 gives access to the data files used in the text, and, for instructors, the remaining solutions. The only pre-requisite for the book 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 useful modern methods such as the bootstrap. This will be a key text for undergraduates in Computer Science, Physics, Mathematics, Chemistry, Biology and Business Studies who are studying a mathematical statistics course, and also for more intensive engineering statistics courses for undergraduates in all engineering subjects.

Author Biography

Michel Dekking, Cor Kraaikamp, Rik Lopuha+ñ and Ludolf Meester are professors in the Department of Applied Mathematics at TU Delft, The Netherlands. The material in this book has been successfully taught there for several years, and at the University of Leiden, The Netherlands, and Wesleyan University, USA, since 2003.

Table of Contents

1 Why probability and statistics?	1	(12)
1.1 Biometry: iris recognition	1	(2)
1.2 Killer football	3	(1)
1.3 Cars and goats: the Monty Hall dilemma	4	(1)
1.4 The space shuttle Challenger	5	(2)
1.5 Statistics versus intelligence agencies	7	(2)
1.6 The speed of light	9	(4)
2 Outcomes, events, and probability	13	(12)
2.1 Sample spaces	13	(1)
2.2 Events	14	(2)
2.3 Probability	16	(2)
2.4 Products of sample spaces	18	(1)
2.5 An infinite sample space	19	(2)
2.6 Solutions to the quick exercises	21	(1)
2.7 Exercises	21	(4)
3 Conditional probability and independence	25	(16)
3.1 Conditional probability	25	(2)
3.2 The multiplication rule	27	(3)
3.3 The law of total probability and Bayes' rule	30	(2)
3.4 Independence	32	(3)
3.5 Solutions to the quick exercises	35	(2)
3.6 Exercises	37	(4)
4 Discrete random variables	41	(16)
4.1 Random variables	41	(2)
4.2 The probability distribution of a discrete random variable	43	(2)
4.3 The Bernoulli and binomial distributions	45	(3)
4.4 The geometric distribution	48	(2)
4.5 Solutions to the quick exercises	50	(1)
4.6 Exercises	51	(6)
5 Continuous random variables	57	(14)
5.1 Probability density functions	57	(3)
5.2 The uniform distribution	60	(1)
5.3 The exponential distribution	61	(2)
5.4 The Pareto distribution	63	(1)
5.5 The normal distribution	64	(1)
5.6 Quantiles	65	(2)
5.7 Solutions to the quick exercises	67	(1)
5.8 Exercises	68	(3)
6 Simulation	71	(18)
6.1 What is simulation?	71	(1)
6.2 Generating realizations of random variables	72	(3)
6.3 Comparing two jury rules	75	(5)
6.4 The single-server queue	80	(4)
6.5 Solutions to the quick exercises	84	(1)
6.6 Exercises	85	(4)
7 Expectation and variance	89	(14)
7.1 Expected values	89	(4)
7.2 Three examples	93	(1)
7.3 The change-of-variable formula	94	(2)
7.4 Variance	96	(3)
7.5 Solutions to the quick exercises	99	(1)
7.6 Exercises	99	(4)
8 Computations with random variables	103	(12)
8.1 Transforming discrete random variables	103	(1)
8.2 Transforming continuous random variables	104	(2)
8.3 Jensen's inequality	106	(2)
8.4 Extremes	108	(2)
8.5 Solutions to the quick exercises	110	(1)
8.6 Exercises	111	(4)
9 Joint distributions and independence	115	(20)
9.1 Joint distributions of discrete random variables	115	(3)
9.2 Joint distributions of continuous random variables	118	(4)
9.3 More than two random variables	122	(2)
9.4 Independent random variables	124	(1)
9.5 Propagation of independence	125	(1)
9.6 Solutions to the quick exercises	126	(1)
9.7 Exercises	127	(8)
10 Covariance and correlation	135	(16)
10.1 Expectation and joint distributions	135	(3)
10.2 Covariance	138	(3)
10.3 The correlation coefficient	141	(2)
10.4 Solutions to the quick exercises	143	(1)
10.5 Exercises	144	(7)
11 More computations with more random variables	151	(16)
11.1 Sums of discrete random variables	151	(3)
11.2 Sums of continuous random variables	154	(5)
11.3 Product and quotient of two random variables	159	(3)
11.4 Solutions to the quick exercises	162	(1)
11.5 Exercises	163	(4)
12 The Poisson process	167	(14)
12.1 Random points	167	(1)
12.2 Taking a closer look at random arrivals	168	(3)
12.3 The one-dimensional Poisson process	171	(2)
12.4 Higher-dimensional Poisson processes	173	(3)
12.5 Solutions to the quick exercises	176	(1)
12.6 Exercises	176	(5)
13 The law of large numbers	181	(14)
13.1 Averages vary less	181	(2)
13.2 Chebyshev's inequality	183	(2)
13.3 The law of large numbers	185	(3)
13.4 Consequences of the law of large numbers	188	(3)
13.5 Solutions to the quick exercises	191	(1)
13.6 Exercises	191	(4)
14 The central limit theorem	195	(12)
14.1 Standardizing averages	195	(4)
14.2 Applications of the central limit theorem	199	(3)
14.3 Solutions to the quick exercises	202	(1)
14.4 Exercises	203	(4)
15 Exploratory data analysis: graphical summaries	207	(24)
15.1 Example: the Old Faithful data	207	(2)
15.2 Histograms	209	(3)
15.3 Kernel density estimates	212	(7)
15.4 The empirical distribution function	219	(2)
15.5 Scatterplot	221	(4)
15.6 Solutions to the quick exercises	225	(1)
15.7 Exercises	226	(5)
16 Exploratory data analysis: numerical summaries	231	(14)
16.1 The center of a dataset	231	(2)
16.2 The amount of variability of a dataset	233	(1)
16.3 Empirical quantiles, quartiles, and the IQR	234	(2)
16.4 The box-and-whisker plot	236	(2)
16.5 Solutions to the quick exercises	238	(2)
16.6 Exercises	240	(5)
17 Basic statistical models	245	(24)
17.1 Random samples and statistical models	245	(3)
17.2 Distribution features and sample statistics	248	(5)
17.3 Estimating features of the "true" distribution	253	(3)
17.4 The linear regression model	256	(3)
17.5 Solutions to the quick exercises	259	(1)
17.6 Exercises	259	(10)
18 The bootstrap	269	(16)
18.1 The bootstrap principle	269	(3)
18.2 The empirical bootstrap	272	(4)
18.3 The parametric bootstrap	276	(3)
18.4 Solutions to the quick exercises	279	(1)
18.5 Exercises	280	(5)
19 Unbiased estimators	285	(14)
19.1 Estimators	285	(2)
19.2 Investigating the behavior of an estimator	287	(1)
19.3 The sampling distribution and unbiasedness	288	(4)
19.4 Unbiased estimators for expectation and variance	292	(2)
19.5 Solutions to the quick exercises	294	(1)
19.6 Exercises	294	(5)
20 Efficiency and mean squared error	299	(14)
20.1 Estimating the number of German tanks	299	(3)
20.2 Variance of an estimator	302	(3)
20.3 Mean squared error	305	(2)
20.4 Solutions to the quick exercises	307	(1)
20.5 Exercises	307	(6)
21 Maximum likelihood	313	(16)
21.1 Why a general principle?	313	(1)
21.2 The maximum likelihood principle	314	(2)
21.3 Likelihood and loglikelihood	316	(5)
21.4 Properties of maximum likelihood estimators	321	(1)
21.5 Solutions to the quick exercises	322	(1)
21.6 Exercises	323	(6)
22 The method of least squares	329	(12)
22.1 Least squares estimation and regression	329	(3)
22.2 Residuals	332	(3)
22.3 Relation with maximum likelihood	335	(1)
22.4 Solutions to the quick exercises	336	(1)
22.5 Exercises	337	(4)
23 Confidence intervals for the mean	341	(20)
23.1 General principle	341	(4)
23.2 Normal data	345	(5)
23.3 Bootstrap confidence intervals	350	(3)
23.4 Large samples	353	(2)
23.5 Solutions to the quick exercises	355	(1)
23.6 Exercises	356	(5)
24 More on confidence intervals	361	(12)
24.1 The probability of success	361	(3)
24.2 Is there a general method?	364	(2)
24.3 One-sided confidence intervals	366	(1)
24.4 Determining the sample size	367	(1)
24.5 Solutions to the quick exercises	368	(1)
24.6 Exercises	369	(4)
25 Testing hypotheses: essentials	373	(10)
25.1 Null hypothesis and test statistic	373	(3)
25.2 Tail probabilities	376	(1)
25.3 Type I and type II errors	377	(2)
25.4 Solutions to the quick exercises	379	(1)
25.5 Exercises	380	(3)
26 Testing hypotheses: elaboration	383	(16)
26.1 Significance level	383	(3)
26.2 Critical region and critical values	386	(4)
26.3 Type II error	390	(2)
26.4 Relation with confidence intervals	392	(1)
26.5 Solutions to the quick exercises	393	(1)
26.6 Exercises	394	(5)
27 The t-test	399	(16)
27.1 Monitoring the production of ball bearings	399	(2)
27.2 The one-sample t-test	401	(4)
27.3 The t-test in a regression setting	405	(4)
27.4 Solutions to the quick exercises	409	(1)
27.5 Exercises	410	(5)
28 Comparing two samples	415	(14)
28.1 Is dry drilling faster than wet drilling?	415	(1)
28.2 Two samples with equal variances	416	(3)
28.3 Two samples with unequal variances	419	(3)
28.4 Large samples	422	(2)
28.5 Solutions to the quick exercises	424	(1)
28.6 Exercises	424	(5)
A Summary of distributions	429	(2)
B Tables of the normal and t-distributions	431	(4)
C Answers to selected exercises	435	(10)
D Full solutions to selected exercises	445	(30)
References	475	(2)
List of symbols	477	(2)
Index	479