Material for Midterm 2

Contents go through Chapter 18 of the textbook, that is, through the lecture on Tuesday 10/29.

Note that the new content since Midterm 1 is in Chapters 10 through 18. However, it is not possible to understand that material without first understanding Chapters 1 through 9.

General Concepts and Methods

Probability

Chapter 1, Lab 1, Lab 2: Spaces, events, basic counting, exponential approximation
Chapter 2: Addition and multiplication rules; conditioning and updating
Chapter 5: Unions and intersections of several events
Section 9.1: Probabilities by conditioning and recursion (discrete)
Section 4.1, 17.1: From joint distributions
Section 4.5, 17.2: Dependence and independence
Section 5.1, 12.3: Bounds – Boole, Markov, Chebyshev

Distribution

Chapter 3: Intro; equality versus equality in distribution
Chapter 4: Joint, marginals, conditionals, independence (discrete case), total variation distance
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, Lab 6: Density of a transformation
Chapter 17: Joint, marginal, and conditional densities; independence
Chapter 14: Distribution of sum
Chapter 14, Section 15.3: Central Limit Theorem

Expectation

Chapter 8, Lab 3: The crucial properties (discrete case) including method of indicators, expectations of functions, tail sum formula (see also geometric distribution)
Section 9.2, 9.3: Expectation by conditioning
Section 15.3, 17.1: Expectation using densities and joint densities

Variance

Chapter 12: Definition and basic properties; linear transformations
Chapter 13, Lab 5: Covariance; variance of a sum (including dependent indicators and simple random sample sums)

Estimation and Prediction

Section 8.4: Unbiased estimators
Section 14.5, 14.6: IID sample mean; confidence interval for population mean
Homework 7: Unbiased estimator of a population variance
Section 12.2: Expectation as a least squares predictor

Models and Special Distributions

Markov Chains

Sections 10.1, 10.2: Terminology and basics
Sections 10.3, 10.4: The steady state distribution and its properties
Section 11.1: Balance and detailed balance
Sections 11.2, 11.3, Lab 4: Code Breaking and MCMC

Random Counts

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, Lab 5, Chapter 7, 8.5, 13.3, 14.3: Binomial and multinomial
Section 5.4, 6.4, 8.5, 13.4: Hypergeometric
Section 5.3, 6.6, Lab 2, Chapter 7, Section 8.2, 8.3, 12.1: Poisson
Section 8.2, 9.3: Geometric

Uniform $(a, b)$

Section 15.3: Density, expectation, variance, CDF
Section 16.3, Lab 6: Use in simulation

Beta

Section 17.4: Integer parameters; uniform order statistics

Normal

Section 14.3, 14.4: CLT; Normal cdf and inverse cdf
Sections 14.6: Normal confidence intervals
Sections 16.1, 18.1: Normal densities, mean, variance, relation with Rayleigh density
Section 18.2: Sums of independent normal variables

Gamma

Section 15.4, 16.1: Exponential and scaling
Sections 18.3.1, 18.3.2, Homework 9: Gamma function, gamma density, mean, variance

Omitted from Scope

Sections 18.3.3, 18.3.4: Sums of independent gamma variables; integer shape parameter
Section 18.4: Sums of squares of iid standard normals; the chi-squared distribution