# Material for Midterm 2

#### A. Adhikari

Contents go through Chapter 17 of the textbook, that is, through the lecture on Tuesday 3/12.

Note that the new content since Midterm 1 is in Chapters 9 through 17. However, it is not possible to understand that material without first understanding Chapters 1 through 8.

## General Concepts and Methods

### Probability

- Chapter 1, Lab 1: 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 â€“ Boole, Markov, Chebyshev

### Distribution

- Chapter 3: Intro; equality versus equality in distribution
- Chapter 4, Lab 2: 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 $1, 2, â€¦, n$
- Section 8.2, 12.1, 13.4: Bernoulli (indicator)
- Section 6.1, 6.2, 6.3, 6.5, Lab 2, 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 \((a, b)\)

### 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
- Section 16.1: Normal densities