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

✍ Scribed by F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester

• Publisher: Springer
• Year: 2005
• Tongue: English
• Leaves: 485
• Series: Springer Texts in Statistics
• Edition: 1
• Category: Library

✦ Table of Contents

Cover
Title page
Copyright page
Preface
Contents
1 Why probability and statistics?
1.1 Biometry: iris recognition
1.2 Killer football
1.3 Cars and goats: the Monty Hall dilemma
1.4 The space shuttle Challenger
1.5 Statistics versus intelligence agencies
1.6 The speed of light
2 Outcomes, events, and probability
2.1 Sample spaces
2.2 Events
2.3 Probability
2.4 Products of sample spaces
2.5 An infinite sample space
2.6 Solutions to the quick exercises
2.7 Exercises
3 Conditional probability and independence
3.1 Conditional probability
3.2 The multiplication rule
3.3 The law of total probability and Bayes’ rule
3.4 Independence
3.5 Solutions to the quick exercises
3.6 Exercises
4 Discrete random variables
4.1 Random variables
4.2 The probability distribution of a discrete random variable
4.3 The Bernoulli and binomial distributions
4.4 The geometric distribution
4.5 Solutions to the quick exercises
4.6 Exercises
5 Continuous random variables
5.1 Probability density functions
5.2 The uniform distribution
5.3 The exponential distribution
5.4 The Pareto distribution
5.5 The normal distribution
5.6 Quantiles
5.7 Solutions to the quick exercises
5.8 Exercises
6 Simulation
6.1 What is simulation?
6.2 Generating realizations of random variables
6.3 Comparing two jury rules
6.4 The single-server queue
6.5 Solutions to the quick exercises
6.6 Exercises
7 Expectation and variance
7.1 Expected values
7.2 Three examples
7.3 The change-of-variable formula
7.4 Variance
7.5 Solutions to the quick exercises
7.6 Exercises
8 Computations with random variables
8.1 Transforming discrete random variables
8.2 Transforming continuous random variables
8.3 Jensen’s inequality
8.4 Extremes
8.5 Solutions to the quick exercises
8.6 Exercises
9 Joint distributions and independence
9.1 Joint distributions of discrete random variables
9.2 Joint distributions of continuous random variables
9.3 More than two random variables
9.4 Independent random variables
9.5 Propagation of independence
9.6 Solutions to the quick exercises
9.7 Exercises
10 Covariance and correlation
10.1 Expectation and joint distributions
10.2 Covariance
10.3 The correlation coefficient
10.4 Solutions to the quick exercises
10.5 Exercises
11 More computations with more random variables
11.1 Sums of discrete random variables
11.2 Sums of continuous random variables
11.3 Product and quotient of two random variables
11.4 Solutions to the quick exercises
11.5 Exercises
12 The Poisson process
12.1 Random points
12.2 Taking a closer look at random arrivals
12.3 The one-dimensional Poisson process
12.4 Higher-dimensional Poisson processes
12.5 Solutions to the quick exercises
12.6 Exercises
13 The law of large numbers
13.1 Averages vary less
13.2 Chebyshev’s inequality
13.3 The law of large numbers
13.4 Consequences of the law of large numbers
13.5 Solutions to the quick exercises
13.6 Exercises
14 The central limit theorem
14.1 Standardizing averages
14.2 Applications of the central limit theorem
14.3 Solutions to the quick exercises
14.4 Exercises
15 Exploratory data analysis: graphical summaries
15.1 Example: the Old Faithful data
15.2 Histograms
15.3 Kernel density estimates
15.4 The empirical distribution function
15.5 Scatterplot
15.6 Solutions to the quick exercises
15.7 Exercises
16 Exploratory data analysis: numerical summaries
16.1 The center of a dataset
16.2 The amount of variability of a dataset
16.3 Empirical quantiles, quartiles, and the IQR
16.4 The box-and-whisker plot
16.5 Solutions to the quick exercises
16.6 Exercises
17 Basic statistical models
17.1 Random samples and statistical models
17.2 Distribution features and sample statistics
17.3 Estimating features of the “true” distribution
17.4 The linear regression model
17.5 Solutions to the quick exercises
17.6 Exercises
18 The bootstrap
18.1 The bootstrap principle
18.2 The empirical bootstrap
18.3 The parametric bootstrap
18.4 Solutions to the quick exercises
18.5 Exercises
19 Unbiased estimators
19.1 Estimators
19.2 Investigating the behavior of an estimator
19.3 The sampling distribution and unbiasedness
19.4 Unbiased estimators for expectation and variance
19.5 Solutions to the quick exercises
19.6 Exercises
20 Efficiency and mean squared error
20.1 Estimating the number of German tanks
20.2 Variance of an estimator
20.3 Mean squared error
20.4 Solutions to the quick exercises
20.5 Exercises
21 Maximum likelihood
21.1 Why a general principle?
21.2 The maximum likelihood principle
21.3 Likelihood and loglikelihood
21.4 Properties of maximum likelihood estimators
21.5 Solutions to the quick exercises
21.6 Exercises
22 The method of least squares
22.1 Least squares estimation and regression
22.2 Residuals
22.3 Relation with maximum likelihood
22.4 Solutions to the quick exercises
22.5 Exercises
23 Confidence intervals for the mean
23.1 General principle
23.2 Normal data
23.3 Bootstrap confidence intervals
23.4 Large samples
23.5 Solutions to the quick exercises
23.6 Exercises
24 More on confidence intervals
24.1 The probability of success
24.2 Is there a general method?
24.3 One-sided confidence intervals
24.4 Determining the sample size
24.5 Solutions to the quick exercises
24.6 Exercises
25 Testing hypotheses: essentials
25.1 Null hypothesis and test statistic
25.2 Tail probabilities
25.3 Type I and type II errors
25.4 Solutions to the quick exercises
25.5 Exercises
26 Testing hypotheses: elaboration
26.1 Significance level
26.2 Critical region and critical values
26.3 Type II error
26.4 Relation with confidence intervals
26.5 Solutions to the quick exercises
26.6 Exercises
27 The t-test
27.1 Monitoring the production of ball bearings
27.2 The one-sample t-test
27.3 The t-test in a regression setting
27.4 Solutions to the quick exercises
27.5 Exercises
28 Comparing two samples
28.1 Is dry drilling faster than wet drilling?
28.2 Two samples with equal variances
28.3 Two samples with unequal variances
28.4 Large samples
28.5 Solutions to the quick exercises
28.6 Exercises
A Summary of distributions
B Tables of the normal and t-distributions
C Answers to selected exercises
D Full solutions to selected exercises
References
List of symbols
Index