Table of Contents
1 Cover
2 Preface
3 1 Data Analysis and Understanding 1.1 Exploring the Distribution of Data 1.2 Exploring Prediction Using Data Problems
4 2 Classical Probability 2.1 Experiments with Equally Likely Outcomes 2.2 Probability Laws 2.3 Counting Methods 2.4 Countable Sets: Implications as 2.5 Kolmogorov's Axioms 2.6 Reliability: Series Versus Parallel Networks Problems
5 3 Random Variables and Models Derived From Classical Probability and Postulates 3.1 Random Variables and Probability Distributions: Discrete Uniform Example 3.2 The Univariate Probability Density Function: Continuous Uniform Example 3.3 Summary Statistics: Central and Non‐Central Moments 3.4 Binomial Experiments 3.5 Waiting Time for a Success: Geometric PMF 3.6 Waiting Time for Successes: Negative Binomial 3.7 Poisson Process and Distribution 3.8 Waiting Time for Poisson Events: Negative Exponential PDF 3.9 The Normal Distribution (Also Known as the Gaussian Distribution) Problems
6 4 Bivariate Random Variables, Transformations, and Simulations 4.1 Bivariate Continuous Random Variables 4.2 Change of Variables 4.3 Simulations Problems
7 5 Approximations and Asymptotics 5.1 Why Do We Like Random Samples? 5.2 Useful Inequalities 5.3 Sequences of Random Variables 5.4 Central Limit Theorem 5.5 Delta Method and Variance‐stabilizing Transformations Problems Notes
8 6 Parameter Estimation 6.1 Desirable Properties of an Estimator 6.2 Moments of the Sample Mean and Variance 6.3 Method of Moments (MoM) 6.4 Sufficient Statistics and Data Compression 6.5 Bayesian Parameter Estimation 6.6 Maximum Likelihood Parameter Estimation 6.7 Information Inequalities and the Cramér–Rao Lower Bound Problems
9 7 Hypothesis Testing 7.1 Setting up a Hypothesis Test 7.2 Best Critical Region for Simple Hypotheses 7.3 Best Critical Region for a Composite Alternative Hypothesis 7.4 Reporting Results: ‐values and Power 7.5 Multiple Testing and the Bonferroni Correction Problems
10 8 Confidence Intervals and Other Hypothesis Tests 8.1 Confidence Intervals 8.2 Hypotheses About the Variance and the ‐Distribution 8.3 Pearson's Chi‐Squared Tests 8.4 Correlation Coefficient Tests and CIs 8.5 Linear Regression 8.6 Analysis of Variance Problems
11
9 Topics in Statistics
9.1 MSE and Histogram Bin Width Selection
9.2 An Optimal Stopping Time Problem
9.3 Compound Random Variables
9.4 Simulation and the Bootstrap
9.5 Multiple Linear Regression
9.6 Experimental