**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)

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

Logic and proofs (P&R 31.1)

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

**Practice problems** (not to turn in): 5.1.3 (a-d); 5.2.3 (d only)

**Homework problems** on sequences and convergence(turn in before class Wednesday 9/4)

## 1. Probability and Statistical Inference

##### September

Text: Excerpts (as shown below) from Wasserman, *All of Statistics*, chapters 1-10

### 1.1 Probability

##### Wed 9/4 though Mon 9/16

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

**Homework problems** to turn in before class Monday 9/9: from Wasserman sec. 1.10,

- problems 5, 13, 21, 22
- Note: problems 21 and 22 are "computer exercises". Please turn in your code as well as the relevant output.

**Homework problems** to turn in before class Monday 9/16: from Wasserman

- Chapter 1: problems 12, 15, 19
- Chapter 2: problems 2, 4, 6

**Homework problems** to turn in before class Monday 9/23: from Wasserman

- Chapter 2: problems 7, 9, 17, 18

### 1.2 Expectations and Moments

Read: sections 3.1-3.5

**Homework problems** to turn in before class Monday 9/30: from Wasserman

- from Chapter 3: problems 3, 4, 5, 11

Finish up Chapter 3 and start Chapter 5

- For Monday 9/30: read section 3.5, 5.1
- Have a look at problem 13 in Chapter 3 -- but you don't need to turn it in

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

### 1.3 Convergence of Random Variables

For Monday 10/7: read section 5.2

For Wednesday 10/9: read section 5.3-5.4

#### (Fall break Mon. 10/14)

**Homework problems** to turn in before class **Wednesday** 10/16 from Wasserman

- from Chapter 5:
- Do the following, which you can label "Exercise A": For each
*n*= 1, 2, ..., let*X*have the uniform distribution on the interval [-1/_{n}*n*, 1/*n*]. Show that*X*converges in probability to 0._{n} - Do the following, which you can label "Exercise B": 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 to_{n}*X*in distribution, butin probability.**not** - In Exercise 4 (p. 83), rather than showing convergence in quadratic mean (you may ignore that part of the question), show that
*X*converges in probability to 0._{n}show__Also,__—that is, relying only on the definition of convergence-in-distrubution, and not on Theorem 5.4(b)—that__directly__*X*converges in distribution to 0._{n}

### 1.4 Statistical Inference and Hypothesis Testing

For Wednesday 10/16 Read: section 6.3

**Homework problems** to turn in before class Monday 10/21

- Chapter 6: exercises 2 and 3.

**Homework problems** to turn in before class Monday 10/28

- Another problem pertaining to Chapter 5. Call this "Exercise C":

For each*n*= 1, 2, ..., let*X*have the following discrete distribution:_{n}

*X*= 0 with probability 1/2 - 1/(_{n}*n*+1) and

*X*= 1 with probability 1/2 + 1/_{n}*(n*+1).

Let*X*be a Bernoulli random variable, equal to 0 or 1 with*p*= 1/2 each.

- Show that
*X*does not converge to_{n}*X*in probability. - Show that
*X*converges to_{n}*X*in distribution.

Readings from Sec. 10.1, hypothesis testing and the Wald test

- For Mon. 10/28 we will cover (and you should read) the remainder of section 6.3 and the Introductory material of Chapter 10.
- For Wed. 10/30: Sec. 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.

**Homework problems** to turn in before class Monday 11/4:

- Call this "Exercise D": use \rnorm to generate a random sample of 20 values all from a normal distribution with mean 10 and variance 4. Now pretend the mean is unknown (but the true variance is still known). Calculate .99, .95, and .9 confidence intervals for the true mean, as estimated by your sample mean.
- from Chapter 10, do 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.)

For Monday 11/4 read remainder of Wasserman topics:

- 10.2
*p*-values - 13.1 linear regression as an application, especially Example 13.6 (p. 211-212)

***** Second Exam out W 11/6; return M 11/11 *****

### 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 Preliminaries: limits, functions, continuity, linear functions, exponential function

**Relevant sections in P & R** (review as needed): 1.1, 3.3, 5.4, 9.1, 9.2

### 2.2 Newton's quotient, differentiation, the Mean Value Theorem, and approximation

**Relevant sections in P & R** (review by Wed. 11/13) 6.1, 10.2-10.4

**Exercises due Mon. 11/18**for Chapters 6, 10

- 6.1.1 on the Newton quotient
- 6.2.3 on differentiability
- 10.1.1 equation of the tangent line
- 10.2.3 l'Hôpital's rule
- 10.3.1 linear and polynomial approximation

### 2.3 Monotone functions; inverse functions

**Review relevant sections in P & R** (see below) for Mon. 12/2

### 2.4 Optimization and convexity

**Review relevant sections in P & R** (see below) for Mon. 12/2

## 3. Linear algebra

#### beginning Mon. 11/18

- 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 & 12 (linear algebra)** -- due Monday 11/25

- linear dependence: 11.1.4
- vector inner product and norm: 13.3.1, 13.3.2, 13.3.7
- matrices as linear mappings: 11.2.3
- systems of equations not RUT following reduction: 12.1.2
- matrix inverse 12.3.2, 12.3.3

**Homework exercises -- due Mon. Dec. 2**

- for Chapters 12 & 13 (linear algebra)
- matrix rank: 12.4.3 (refers to 11.1.4)
- determinant: 13.1.1

- SAVE FOR LATER: quadratic forms: 13.4.7
- ALSO see below: review exercises on univariate calculus

### Some useful online text material for additional help

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). - (Best from our point of view to skip chapters 1 and 3.)

Four pages of clear, simple notes on "Finding Eigenvalues And Eigenvectors" from Prof. Rozenn Dahyot of Trinity College Dublin are also posted online

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) is posted on the MIT website.

Another alternative, and interesting, definition of the determinant, in terms of permutations of the columns is given here. Unfortunately one can't tell who is responsible for it.

### Single-variable calculus review

#### Monotone functions; inverse functions

- Review in P & R: sections 7.2-7.4

#### Optimization and convexity

- Review in P & R: Chapter 8

**Exercises for Chapters 7, 8 for Mon. Dec. 2**

- TURN IN Monday Dec. 2: 8.4.2, 8.4.3, 8.4.4 on concavity & convexity
- NOT to turn in, but bring questions to class Mon. Dec. 2
- 7.2.2 composite function rule
- 7.3.1 monotone functions
- 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
- 8.4.1 more practice on concavity and convexity

### (Thanksgiving break W 11/27)

#### Save for later:

### 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

## 4. Multivariate calculus

#### dates TBA

material from Chapters 14, 15, 16

- quadratic forms: 13.4.7
- 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)** -- not to hand in, but bring questions to help session Dec. 6.

- 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. - global optima, concavity, and convexity: 16.2.1 (for a, b, c only); 16.2.2

A nice, free text on multivariate optimization is Martin Osborne's *Mathematical Methods for Economic Theory*. It begins with a simple review of calculus and of vectors and matrices and, later, includes a nice treatment on quadratic forms.

**Third Exam**

- Fri. 12/6 regular help session; Tue. 12/10 optional review session
- Exam distributed by Wed. 12/11
- Exam due M 12/16