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

Description

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