Preamble
1
Axioms of Probability Theory
1.1
Manipulation of Sets
1.2
Venn and Euler diagrams
2
Discrete Probability Spaces
2.1
Bernoulli trials
2.2
Sampling without replacement
2.3
Pólya’s urn model
2.4
Factorials and binomials coefficients
3
Discrete Distributions
3.1
Uniform distributions
3.2
Binomial distributions
3.3
Geometric distributions
3.4
Poisson distributions
4
Distributions on the Real Line
5
Continuous Distributions
5.1
Uniform distributions
5.2
Normal distributions
5.3
Exponential distributions
5.4
Normal approximation to the binomial
6
Multivariate Distributions
6.1
Multinomial distribution
6.2
Uniform distributions
6.3
Normal distributions
7
Expectation and Concentration
7.1
Concentration inequalities
8
Convergence of Random Variables
8.1
Law of Large Numbers
8.2
Central Limit Theorem
8.3
Extreme Value Theory
9
Stochastic Processes
9.1
Poisson processes
9.2
Markov chains
9.3
Simple random walk
9.4
Galton–Watson processes
9.5
Random graph models
9.5.1
Gilbert and Erdos–Renyi random graphs
9.5.2
Percolation models
9.5.3
Random geometric graph
10
Sampling and Simulation
10.1
Monte Carlo simulations
10.2
Monte Carlo integration
10.3
Markov chain Monte Carlo
11
Data Collection
11.1
Survey sampling
11.1.1
Sampling according to prescribed inclusion probabilities
11.1.2
Cluster sampling
11.2
Experimental design
11.3
Observational studies
12
Models, Estimators, and Tests
13
Properties of Estimators and Tests
13.1
Comparing estimators
13.2
Uniformly most powerful tests
14
One Proportion
14.1
Comparing various confidence intervals
15
Multiple Proportions
15.1
One-sample goodness-of-fit testing
15.2
Association studies
15.3
Matched-pair experiment
16
One Numerical Sample
16.1
Quantiles and other summary statistics
16.2
Empirical distribution function
16.3
Histogram
16.4
Bootstrap confidence interval
16.5
Goodness-of-fit testing
16.6
Kaplan–Meier estimator
17
Multiple Numerical Samples
17.1
An example of A/B testing
17.2
Plots
17.2.1
Histograms
17.2.2
Violin plots
17.3
Tests
18
Paired Numerical Samples
18.1
Scatterplot
18.2
Testing for symmetry
19
Correlation Analysis
19.1
Sample correlations
19.2
Correlations tests
19.3
Distance covariance (and test)
20
Multiple Testing
20.1
An example from genetics
20.1.1
Adjusted p-values
20.1.2
Rejections
20.2
An example from meta-analysis
20.2.1
Cochran–Mantel–Haenszel test
20.2.2
Combination tests
21
Regression Analysis
21.1
Regression
21.1.1
Kernel smoothing
21.1.2
Local linear regression
21.1.3
Polynomial regression
21.2
Classification
21.2.1
Nearest neighbor classifier
Principles of Statistical Analysis: R Companion
4
Distributions on the Real Line