Chapter 1 , Lab 1, Lab 2: Spaces, events, basic counting, exponential approximationChapter 2 : Addition and multiplication rules; conditioning and updatingChapter 5 : Unions and intersections of several events; exact probabilities and boundsSection 9.1 : Probabilities by conditioning and recursion (discrete)
Section 20.2 : Probabilities by conditioning on a continuous variable
Section 4.5 , 20.2 : Independence
Chapter 3 : Intro; equality versus equality in distributionChapter 4 : Joint, marginals, conditionals, independence (discrete case)Section 5.3 , 5.4 : Random permutations and symmetry
Section 15.1 , 15.2 , Lab 6: Density
Section 6.1 , 15.1 , 16.3 , Lab 6: CDF and inverse CDF
Chapter 16 , Section 23.3 , Lab 6, Lab 8: Density of a transformationChapter 17 : Joint, marginal, and conditional densities; independenceChapter 14 , Chapter 19 : Distribution of sumChapter 14 , Section 15.3 , 19.3 : Central Limit Theorem
Chapter 8 , Lab 3B: The crucial properties (discrete case) including method of indicators, expectations of functions, tail sum formula (see also geometric distribution)Section 12.3 , 19.4 : Tail bounds: Markov, Chebyshev, Chernoff
Section 9.2 , 9.3 : Expectation by conditioning
Section 15.3 , 17.1 , 20.2 : Expectation using densities and joint densities, and by conditioning on a continuous variable
Section 19.2 : Moment generating function
Chapter 12 : Definition and basic properties; linear transformationsChapter 13 , Lab 5: Covariance; variance of a sumSection 24.2 , Homework 7, Lab 5: Correlation and its properties
Section 22.3 , 22.4 : Variance by conditioning, mixtures
Section 23.1 : Mean vector and covariance matrix of a random vector; linear transformations
Section 8.4 : Unbiased estimators
Section 14.5 , 14.6 : IID sample mean; confidence interval for population mean
Homework 7, Homework 15: Unbiased estimator of a population variance; independence of normal sample mean and sample variance
Section 20.1 : Maximum likelihood estimate
Section 20.3 : Posterior density, MAP estimate
Section 12.2 , 22.1 , 22.2 : Expectation and conditional expectation as least squares predictors
Section 24.1 , 25.4 : Least squares linear predictor
Section 8.1 , 12.1 : Uniform on \(1, 2, ..., n\)
Section 8.2 , 12.1 , 13.4 : Bernoulli (indicator)
Section 6.1 , 6.2 , 6.3 , 6.5 , Chapter 7 , 8.5 , 13.3 , Lab 5, 14.3 , 19.2 , Chapter 21 : Binomial and multinomial
Section 5.4 , 6.4 , 8.5 , 13.4 : Hypergeometric
Section 5.3 , 6.6 , Chapter 7 , Section 8.2 , 8.3 , 12.1 , 19.2 , Lab 7: Poisson
Section 8.2 , 9.3 , 22.4 : Geometric
Section 15.3 : Density, expectation, variance, CDF
Section 16.3 , Lab 6: Use in simulation
Section 19.1 : Density of sum
Section 14.3 , 14.4 : CLT; Normal cdf and inverse cdf
Sections 14.6 : Normal confidence intervals
Section 16.1 : Normal densities
Section 18.1 , 18.2 , 18.4 : Independent normal variables, linear combinations, squares, Rayleigh, chi-squared
Section 19.3 : Normal MGF, sums, CLT
Section 24.2 , 24.3 , Lab 8: Bivariate normal, linear combinations, independence, regression
Chapter 23 , Section 25.4 : Multivariate normal, linear combinations, independence, regression
Section 15.4 , 16.1 ,
16.2.3 ,
18.1 : Exponential and scaling; square root and the Rayleigh
Homework 9: Gamma function, gamma density, mean, variance
Section 18.3 , 18.4 : Gamma and scaling; chi-squared
Section 19.2 : Sums of independent gammas with the same rate
Lab 7: Waiting times of arrivals in a Poisson process
Homework 15: The chi-squared and the normal sample variance
Section 5.2 (general inclusion-exclusion formula)
Chapters 10, 11 (Markov Chains)
Section 12.4 (Heavy-tailed distributions)
Sections 14.1, 14.2 (Probability generating functions)
Section 16.4.1 (Two-to-one function change of variable for densities)
Section 19.3.4 (“Proof” of the Central Limit Theorem)
Section 21.3 (Long-run proportion of heads for a random coin)
Sections 25.1, 25.2, 25.3 (general best linear predictor based on multiple predictors)