**Fall 2019**

This course is an introduction to mathematical techniques used to model phenomena studied in political science, with special attention to the analysis of individual action. Mathematical topics covered include: sets, functions, and graphs; matrix algebra; differential calculus and optimization; probability, mathematical statistics, and decisions under risk; integral calculus; and sequences, series, and limits. All these topics are useful in many settings in political science, including game theory, dynamic modeling, and statistics.

This course website will be updated to reflect any changes in schedule, topics covered, or assignments, as well as to provide relevant links to materials associated with the course.

## Course Outline and Approximate Schedule

## 0. Some Preliminaries

##### Monday and Wednesday, 8/26-8/28

Sets, real numbers, and intervals (reference: Pemberton & Rao sections 3.1-3.2; 31.2)

Logic and proofs (P&R 31.1)

Sequences, convergence, and limits (P&R 5.1, 5.4; for advanced help, 31.3, 31.4)

- Exercises on sequences and convergence: prepare your answers using LaTeX and bring them to class next time.

Areas, sums, and integrals--brief review if needed (P&R 19.1)

## 1. Probability and Statistical Inference

Text: Excerpts from Wasserman, *All of Statistics*, chapters 1-10

**Exercises to be assigned** (subject to revision); due dates TBA

- Chapter 1: Exercises 5, 12, 13, 15, 19; Computer Experiments 21, 22
- Chapter 2: Exercises 2, 4, 6, 7, 9, 17, 18
- Chapter 3: Exercises 3, 4, 5, 11, 13
- Chapter 5:
- Do the following practice problem, which you can label "Problem A": For each
*n*= 1, 2, ..., let*X*have the uniform distribution on the interval [-1/_{n}*n*, 1 + 1/*n*]. Let*X*be distributed uniformly on [0, 1]. Show that*X*converges both in probability and in distribution to_{n}*X*. (You should prove convergence in distribution__directly__-- that is, without appealing to the fact that convergence in probability implies convergence in distribution.) - Do the version of Exercise 4 as we discussed in class: Don't examine convergence in "quadratic mean." Rather, in addition to showing that
*X*converges in probability to 0, show (_{n}__directly__, again) that it also converges in distribution to 0.

- Do the following practice problem, which you can label "Problem A": For each
- Chapter 6: Exercise 2, 3
- Chapter 10: Exercise 6 (Hint:
*n*= 1919 is definitely a "large" sample, so the binomial distribution suggested in the exercise is approximately normal. Use the Wald test.)

### 1.1 Probability

##### Wed 9/4, Mon. 9/9

**For Wednesday 9/4 read: Wasserman, Chapters 1 and 2.1-2.2**

### 1.2 Expectations and Moments

Read: sections 3.1-3.5

### 1.3 Convergence of Random Variables

Read: sections 5.1-5.4

### 1.4 Statistical Inference and Hypothesis Testing

Read: sections 6.3, 10.1, 10.2, 13.1

- 6.3 estimation, confidence sets, and hypothesis testing
- linear regression as an application: Example 6.6 (p. 89)

- 10.1 the Wald test: read only through Remark 10.5, plus Example 10.8 (Comparing Two Means) and Theorem 10.10 and the Warning following it.
- 10.2
*p*-values- linear regression as an application: Section 13.1, especially Example 13.6 (p. 211-212)

***** First Exam out about W 10/2, in M 10/7 *****

#### (Fall break M 10/14)

### For all the subjects below, chapter and section numbers refer to Pemberton & Rau, *Mathematics for Economists* (4th edition)

## 2. Calculus of one variable

### 2.1 Newton's quotient and differentiation

Read Chapters 6 and 7 (excluding 6.4)

particularly including

- linear approximation using Newton's quotient
- composite functions
- monotone functions
- inverse functions

**Exercises for Chapters 6 and 7** -- due dates TBA

- 6.1.1
- 6.2.1 and 6.2.3
- 7.2.2 and 7.2.3, all parts
- 7.3.1, all parts
- 7.4.1

***** Second Exam out about W 10/30 in M 11/4 *****

### 2.2 Optimization and convexity

Chapter 8

### 2.3 The Exponential and Logarithmic Functions

(Review Sections 4.3, 4.4.) Read Sections 9.1, 9.2

### 2.4 Mean value theorem, l'Hôpital's rule, Taylor's theorem

Sections 10.2 through 10.4

**Exercises for Chapters 8, 9, 10 (differential calculus)** -- due dates TBA

- 8.2.2, 8.2.3, 8.2.4 finding and assessing critical points, using the second derivative; and curve-sketching
- refers back to 8.1.2, 8.1.3

- 8.3.1, 8.3.2 global max & min
- 8.4.2, 8.4.3, 8.4.4 concavity, convexity.
- For optional extra practice do 8.4.1 (don't turn in).

- 9.1.4
- 9.2.2, 9.2.4, 9.2.5
- 10.2.3 all parts (a)-(c)
- 10.2.1, 10.3.2; 10.4.1 (a) and (b)

### 2.5 Integration

Chapters 19, 20 (excluding 19.3)

**Exercises for Chapters 19, 20 (integral calculus)** -- due dates TBA

- area: 19.1.3
- integration: 19.2.3, 19.2.4 (all 3 parts); 19.2.6
- integration by parts & by substitution: 20.1.1, 20.1.3, 20.1.4, 20.1.5
- improper integrals: 20.2.1

### (Thanksgiving break W 11/27)

## 3. Linear algebra

- vectors, linear dependence, length: Section 11.1, 13.3
- matrices as linear mappings; inverses: Section 11.2, 12.3
- linear dependence and rank: Section 12.4
- determinants and quadratic forms: Chapter 13
- eigenvalues and eigenvectors: Chapter 27

**Exercises for Chapters 11-13 (linear algebra)** -- due dates TBA

- vector arithmetic: 11.1.4
- vector length; norm: 13.3.1, 13.3.2, 13.3.7
- matrices as linear mappings: 11.2.3
- matrix rank: 12.4.3 (refers to 11.1.4)
- determinant: 13.1.1
- quadratic forms: 13.4.7

**A nice supplemental FREE linear algebra textbook:** Pemberton and Rau occasionally make the connection between matrix operations and the corresponding properties of linear transformations, but their treatment of linear transformations is not complete. For your future reference, a nice-looking accessible supplementary treatment can be found online:

- David Cherney, Tom Denton, Rohit Thomas, and Andrew Watson,
*Linear Algebra*, online (1.4 megabytes, 460 pages), University of California, Davis (2013).

We have covered material corresponding to some of that in Chapters 1, 2, 4-12, and 16.

A succinct statement of the **test for positive and negative (semi-)definiteness of a symmetric matrix** based on its principal minors (by economist Nathan Barczi) can be found at MIT's website.

Another alternative, and interesting, definition of the determinant, in terms of permutations of the columns is given here.

## 4. Multivariate calculus

material from Chapters 14, 15, 16

- partial derivatives: Read Section 14.1 up through p. 271 only.
- the chain rule: Read Section 14.2, especially begining from top of p. 276
- if time: implicit functions, section 15.1
- optimization: Read sections 16.1, 16.2

**Exercises for Chapters 14-16 (multivariate calculus)** -- due dates TBA

- critical points and second-order conditions: 16.1.2 (a, b, c), 16.1.3 (a, b, c), 16.1.5
- Hint for 16.1.3: consider movements along the line
*y*=*kx*for any constant*k*(whether positive or negative); then- in (a),
*f*(*x*,*y*) =*x*^{4}+*k*^{4}*x*^{4}= (1 +*k*^{4})*x*^{4}; - In (c), it becomes (1 -
*k*^{4})*x*^{4}, and whether*f*reaches a max or min at (0, 0) depends on whether*k*is greater than or less than 1.

- in (a),

- Hint for 16.1.3: consider movements along the line
- global optima, concavity, and convexity: 16.2.1 (for a, b, c only); 16.2.2

**Third Exam**

- Fri. 12/6 regular help session; Mon. 12/9 optional review session
- Exam distributed by Wed. 12/11
- Exam due M 12/16