1.1 Reality, Nature, Science, and Models
1.2 Statistical Processes: Nature, Design and Measurement, and Data
1.3 Models
1.4 Deterministic Models
1.5 Variability
1.6 Parameters
1.7 Purely Probabilistic Statistical Models
1.8 Statistical Models with Both Deterministic and Probabilistic Components
1.9 Statistical Inference
1.10 Good and Bad Models
1.11 Uses of Probability Models
Vocabulary and Formula Summaries
Exercises

2. Random Variables and Their Probability Distributions
2.1 Introduction
2.2 Types of Random Variables: Nominal, Ordinal, and Continuous
2.3 Discrete Probability Distribution Functions
2.4 Continuous Probability Distribution Functions
2.5 Some Calculus: Derivatives and Least Squares
2.6 More Calculus: Integrals and Cumulative Distribution Functions
Vocabulary and Formula Summaries
Exercises

3. Probability Calculation and Simulation
3.1 Introduction
3.2 Analytic Calculations, Discrete and Continuous Cases
3.3 Simulation-Based Approximation
3.4 Generating Random Numbers
Vocabulary and Formula Summaries
Exercises

4. Identifying Distributions
4.1 Introduction
4.2 Identifying Distributions from Theory Alone
4.3 Using Data: Estimating Distributions via the Histogram
4.4 Quantiles: Theoretical and Data-Based Estimates
4.5 Using Data: Comparing Distributions via the Quantile–Quantile Plot
4.6 Effect of Randomness on Histograms and q–q Plots
Vocabulary and Formula Summaries
Exercises

5. Conditional Distributions and Independence
5.1 Introduction
5.2 Conditional Discrete Distributions
5.3 Estimating Conditional Discrete Distributions
5.4 Conditional Continuous Distributions
5.5 Estimating Conditional Continuous Distributions
5.6 Independence
Vocabulary and Formula Summaries
Exercises

6. Marginal Distributions, Joint Distributions, Independence, and Bayes’ Theorem
6.1 Introduction
6.2 Joint and Marginal Distributions
6.3 Estimating and Visualizing Joint Distributions
6.4 Conditional Distributions from Joint Distributions
6.5 Joint Distributions When Variables Are Independent
6.6 Bayes’ Theorem
Vocabulary and Formula Summaries
Exercises

7. Sampling from Populations and Processes
7.1 Introduction
7.2 Sampling from Populations
7.3 Critique of the Population Interpretation of Probability Models
7.4 Sampling from Processes and Populations
7.5 Independent and Identically Distributed Random Variables and Other Models
7.6 Checking the iid Assumption
Vocabulary and Formula Summaries
Exercises

8. Expected Value and the Law of Large Numbers
8.1 Introduction
8.2 Discrete Case
8.3 Continuous Case
8.4 Law of Large Numbers
8.5 Law of Large Numbers for the Bernoulli Distribution
8.6 Keeping the Terminology Straight: Mean, Average, Sample Mean, Sample Average, and Expected Value
8.7 Bootstrap Distribution and the Plug-In Principle
Vocabulary and Formula Summaries
Exercises

9. Functions of Random Variables: Their Distributions and Expected Values
9.1 Introduction
9.2 Distributions of Functions: The Discrete Case
9.3 Distributions of Functions: The Continuous Case
9.4 Expected Values of Functions and the Law of the Unconscious Statistician
9.5 Linearity and Additivity Properties
9.6 Nonlinear Functions and Jensen’s Inequality
9.7 Variance
9.8 Standard Deviation, Mean Absolute Deviation, and Chebyshev’s Inequality
9.9 Linearity Property of Variance
9.10 Skewness and Kurtosis
Vocabulary and Formula Summaries
Exercises

10. Distributions of Totals
10.1 Introduction
10.2 Additivity Property of Variance
10.3 Covariance and Correlation
10.4 Central Limit Theorem
Vocabulary and Formula Summaries
Exercises

11. Estimation: Unbiasedness, Consistency, and Efficiency
11.1 Introduction
11.2 Biased and Unbiased Estimators
11.3 Bias of the Plug-In Estimator of Variance
11.4 Removing the Bias of the Plug-In Estimator of Variance
11.5 The Joke Is on Us: The Standard Deviation Estimator Is Biased After All
11.6 Consistency of Estimators
11.7 Efficiency of Estimators
Vocabulary and Formula Summaries
Exercises

12. Likelihood Function and Maximum Likelihood Estimates
12.1 Introduction
12.2 Likelihood Function
12.3 Maximum Likelihood Estimates
12.4 Wald Standard Error
Vocabulary and Formula Summaries
Exercises

13. Bayesian Statistics
13.1 Introduction: Play a Game with Hans!
13.2 Prior Information and Posterior Knowledge
13.3 Case of the Unknown Survey
13.4 Bayesian Statistics: The Overview
13.5 Bayesian Analysis of the Bernoulli Parameter
13.6 Bayesian Analysis Using Simulation
13.7 What Good Is Bayes?
Vocabulary and Formula Summaries
Exercises

14. Frequentist Statistical Methods
14.1 Introduction
14.2 Large-Sample Approximate Frequentist Confidence Interval for the Process Mean
14.3 What Does Approximate Really Mean for an Interval Range?
14.4 Comparing the Bayesian and Frequentist Paradigms
Vocabulary and Formula Summaries
Exercises

15. Are Your Results Explainable by Chance Alone?
15.1 Introduction
15.2 What Does by Chance Alone Mean?
15.3 The p-Value
15.4 The Extremely Ugly “pv ≤ 0.05” Rule of Thumb
Vocabulary and Formula Summaries
Exercises

16. Chi-Squared, Student’s t, and F-Distributions, with Applications
16.1 Introduction
16.2 Linearity and Additivity Properties of the Normal Distribution
16.3 Effect of Using an Estimate of σ
16.4 Chi-Squared Distribution
16.5 Frequentist Confidence Interval for σ
16.6 Student’s t-Distribution
16.7 Comparing Two Independent Samples Using a Confidence Interval
16.8 Comparing Two Independent Homoscedastic Normal Samples via Hypothesis Testing
16.9 F-Distribution and ANOVA Test
16.10 F-Distribution and Comparing Variances of Two Independent Groups
Vocabulary and Formula Summaries
Exercises

17. Likelihood Ratio Tests
17.1 Introduction
17.2 Likelihood Ratio Method for Constructing Test Statistics
17.3 Evaluating the Statistical Significance of Likelihood Ratio Test Statistics
17.4 Likelihood Ratio Goodness-of-Fit Tests
17.5 Cross-Classification Frequency Tables and Tests of Independence
17.6 Comparing Non-Nested Models via the AIC Statistic
Vocabulary and Formula Summaries
Exercises

18. Sample Size and Power
18.1 Introduction
18.2 Choosing a Sample Size for a Prespecified Accuracy Margin
18.3 Power
18.4 Noncentral Distributions
18.5 Choosing a Sample Size for Prespecified Power
18.6 Post Hoc Power: A Useless Statistic
Vocabulary and Formula Summaries
Exercises

19. Robustness and Nonparametric Methods
19.1 Introduction
19.2 Nonparametric Tests Based on the Rank Transformation
19.3 Randomization Tests
19.4 Level and Power Robustness
19.5 Bootstrap Percentile-t Confidence Interval
Vocabulary and Formula Summaries
Exercises

20. Final Words