PS 5052 Mathematical Modeling in Political Science

Fall 2017

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, 8/28-8/30

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 (P&R 19.1)

 

1. Probability and Statistical Inference

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

1.1 Probability

W 8/30; W 9/6

For Wednesday 9/6 read: chapters 1 and 2.1-2.2

Exercises to turn in Monday 9/11: Chapter 1, exercises 5, 12, 13, 15.

Exercises to turn in Monday 9/18:

  • Chapter 1, Exercise 19; Computer Experiments 21, 22
  • Chapter 2:  Exercises 2, 6

Exercises to turn in Monday 9/25 (subject to adjustment):

  • Chapter 2:  Exercises 4, 7, 9, 18

 

1.2 Expectations and Moments

F 9/29- M 10/2

Read: sections 3.1-3.5

Exercises to turn in Monday 10/2 (subject to adjustment):

  • Chapter 2:  Exercise 17
  • Chapter 3:  Exercises 3, 4, 5, 11, 13

 

First Exam 10/4 - 10/9

  • Wed. 10/4:  Review session
  • Wed. or Fri.:  distribute exam
  • Exam due at beginning of class, Mon. 10/9

 

 

1.3 Convergence of Random Variables

M 10/9

Read: sections 5.1-5.4

No homework will be due on Monday 10/16. Exercises to turn in Monday 10/23:

  • Chapter 5, "Problem A" and a version of Exercise 4, both as described below under "Exercises to be assigned."
  • Exercises from Chapters 6 and 10, TBA.

 

1.4 Statistical Inference and Hypothesis Testing

W 10/11; W 10/18; M 10/23

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)

 

Exercises to turn in Monday 10/23:

  • Chapter 5, "Problem A" and a version of Exercise 4, both as described below under "Exercises to be assigned."
  • Chapter 6: Exercise 2, 3
  • Chapter 10: Exercise 6 (see hint below)

 

Exercises to be assigned (subject to revision)

  • 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 Xn have the uniform distribution on the interval [-1/n, 1 + 1/n]. Let X be distributed uniformly on [0, 1]. Show that Xn converges both in probability and in distribution to 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 Xn converges in probability to 0, show (directly, again) that it also converges in distribution to 0.
  • 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.)

 

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

For Wednesday 10/25 read Chapters 6 and 7 (excluding 6.4)

particularly including

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

Exercises to turn in Monday 10/30

  • 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 11/1 in M 11/6 ***

 

 

2.2 Optimization and convexity

(M 11/6 - W 11/8) Chapter 8

 

2.3 The Exponential and Logarithmic Functions

(W 11/8) (Review Sections 4.3, 4.4.) Read Sections 9.1, 9.2

 

Exercises to turn in Monday 11/13

  • all exercises from Master List below from Chapter 8
  • exercises from Chapter 9 from Master List covered in class W 11/8 (to be announced in class)
  • any exercises from Chapter 10 from Master List covered in class W 11/8 (10.2.3)

 

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

(M 11/13) Sections 10.2 through 10.4

 

Master List of exercises to be assigned from Chapters 8, 9, 10 (differential calculus)

  • 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

(W 11/15) Chapters 19, 20 (excluding 19.3)

Exercises to turn in Monday 11/20

  • Exercises from chapter 10 (Master List, above) not yet done, namely: 10.2.1, 10.3.2; 10.4.1 (a) and (b)
  • Exercise 19.1.3

Master List of exercises to be assigned from Chapters 19, 20 (integral calculus)

  • 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 11/23)

 

3. Linear algebra

(M 11/21; M 11/27 & W 11/29)

  • vectors; matrices as linear mappings: Section 11.1, 11.2
  • linear dependence and rank: Section 12.4
  • determinants and quadratic forms: Chapter 14
  • (if time:) eigenvalues and eigenvectors: Chapter 27

 

4. Multivariate calculus

(M 12/4) material from chapters 14, 15, 16

  • partial derivatives, gradient
  • the chain rule
  • implicit functions
  • optimization

 

5. Analysis

(as time allows -- W 12/6?)

  • continuity and compactness
  • fixed points
  • real numbers
  • open sets and closed sets
  • distance and measure

 

 

Third Exam following 12/6

  • Fri. 12/8 or M 12/11 optional review session
  • within a day after review session:  distribute exam
  • Exam due F 12/15

 


 

 

This page written by Randall Calvert  2017
Email comments and questions to calvert at wustl.edu
Monday & Wednesday 10:00-11:30
classroom: Seigle 104

Jump directly to CURRENT topics & assignments

Assistant to Instructor
Ryden Butler
WU email address r.butler
Office Hours:  TBA

Optional Help Sessions: Fridays 10:00-11:30 in Seigle 104

Textbooks

The following are available in the campus store. We will use them extensively, and they will be useful to you as reference books once the course is finished.

  • Malcolm Pemberton and Nicholas Rau, Mathematics For Economists: An Introductory Textbook, 4th ed. (Manchester University Press, 2016). Answers to "exercises" and to "problems" in Pemberton and Rau. along with errata for the book, are available online at the textbook website.
  • Larry Wasserman, All of Statistics (Springer, 2004). Wasserman's textbook website offers data and rough R code for some exercises and examples.

Course Requirements

All assignments are to be turned in as LaTeX documents; hand-drawn illustrations OK.

  • Exams 75%.  Three, non-cumulative, closed-book, take-home, un-timed. Dates are indicated in the course outline; I will give at least two weeks warning of any changes in exam dates.
  • Homework 25%.  Problem sets drawn mostly from the textbooks; due every Monday, with few exceptions. Collaboration is encouraged. Grading will be based heavily on (1) completion and (2) effort.
  • Possible short quizzes.  I retain the option of administering a short quiz from time to time. These would be graded more for accuracy than are the homeworks, and the scores will be counted in with the 25% for homework.
  • Attendance.  Provided you're not contagious, I expect your attendance at every class meeting. Please let me know, in advance if possible, if problems arise that will require you to be absent.