PS 5052 Mathematical Modeling in Political Science

Fall 2018

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; linear 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, across formal theory 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.


Jump directly to CURRENT topics & assignments

Course Outline and Approximate Schedule

0. Some Preliminaries

Monday and Wednesday, 8/27-8/29

Sets and functions (Pemberton & Rao Chapter 3)

Intervals and real numbers (reference: P&R section 31.2)

A word on logic and proofs (P&R 31.1)

Sequences, convergence, limits, and continuity (P&R Chapter 5; for advanced reference, 31.3 & 31.4)

Exercises to discuss on Friday

  • Try proving the other of DeMorgan's Laws.
  • 5.1.3
  • If we get to series on Wednesday: 5.2.2 and 5.2.3
  • 5.4.1, 5.4.2, 5.4.3
  • Additional exercises on sequences and convergence.
  • If we get to Proposition 3 from p. 714 by Wednesday: 31.4.2. Erratum: Rather than using property SQ8 as the exercise says, they mean to invite you to use Proposition 3. If you do that, you won't need to invoke any epsilons or deltas or Ns to prove the claim in the exercise.

Homework to turn in Wed. Sept. 5

  • As a LaTeX exercise: Exercise 5.4.3 (b)

Exercises to discuss on Friday Sept. 7

  • left over from Chapter 5 and related material on functions and continuity, series
    • 5.2.2 and 5.2.3
    • 5.4.1, 5.4.2, 5.4.3 (including LaTeX for 5.4.3(b))
    • 31.4.2. Erratum: Rather than using property SQ8 as the exercise says, they mean to invite you to use Proposition 3. If you do that, you won't need to invoke any epsilons or deltas or Ns to prove the claim in the exercise.

Homework to turn in Monday Sept. 10

  • Do a new version of problem 5.2.2 and 5.2.3—call it part (e)—namely, a geometric progression with first term a = 1 and common ratio r = .95.



1. Calculus of one variable

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 to discuss on Friday Sept. 14

  • from Chapter 6 on differentiation
    • 6.1.1
    • 6.2.1: In each case, in addition to finding the approximate change in f(x) as instructed in the textbook, also find the exact change, show how much they differ, and compare that to the size of h, as suggested at the end of the first paragraph on p. 109.
    • 6.2.3
    • Additional exercise: Let y = f(x) = x8 - x7 + x6 - x5 + x4. Using linear approximation, find f(1.02). How far off is your approximation from the true value?
  • from Chapter 7 on methods of differentiation: any of the following that we get to by Wednesday in class, as announced
    • 7.2.2 and 7.2.3, all parts

Homework to turn in Monday Sept. 17

  • 6.1.1
  • 7.2.2 (g) only

Exercises to discuss on Friday Sept. 21

  • any remaining problems from Chapter 7 on methods of differentiation
    • 7.3.1, all parts
    • 7.4.1
  • see below for additional exercises from Chapter 8


Optimization and convexity

Chapter 8 Maxima and Minima -- read the whole chapter, even the economics application on pp. 150-51.

More exercises to discuss on Friday Sept. 21

  • from Chapter 8 on maxima and minima
    • 8.1.2, 8.1.3 critical points and curve-sketching
    • 8.2.2, 8.2.3, 8.2.4 using second derivatives
    • 8.3.1, 8.3.2, 8.3.4, 8.3.6 on global max and min
    • 8.4.2, 8.4.3, 8.4.4 concave and convex functions

Homework to turn in Monday Sept. 24

  • 7.4.1 with respect to the function in 7.3.1 (b) only
  • 8.3.6


*** First Exam out about Wed. Sep. 26 ***


The Exponential and Logarithmic Functions

For Monday Oct. 1: review Sections 4.3, 4.4; read Sections 9.1, 9.2


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

For Wed. Oct. 3: read Sections 10.2 through 10.4

Exercises to discuss Friday Oct. 5

  • 9.1.4
  • 9.2.2, 9.2.5
  • 10.2.3 all parts (a)-(c)
  • 10.3.2 10.4.1 (a)

Homework to turn in Monday Oct. 8

  • 9.2.4
  • 10.4.1 (b)



For Oct. 8 & 10, read Chapters 19, 20 (excluding 19.3)

Exercises to discuss Friday, Oct. 12

  • 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(a), 20.1.3, 20.1.4, 20.1.5

Homework to turn in Monday Oct. 15

  • 20.1.1(b)

Exercises to discuss Fri. Oct. 19

  • improper integrals: 20.2.1
  • plus any preliminaries on linear algebra we might cover on Wednesday


2. Linear algebra

Two weeks, Mon. Oct. 22 through Fri. Nov. 2.

We'll be making extensive use of matrix arithmetic (adddition and multiplication) and Gaussian elimination to solve systems of linear equations; these are techniques I assume you've already learned, but we'll slow down as needed. Ask questions. For reference, consult sections 11.2 through 11.4 and 12.1-12.2.

  • 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 discussed Friday Oct. 19.

  • vector arithmetic: 11.1.4

Exercises on linear algebra to discuss Friday Oct. 26.

  • vector length; norm: 13.3.1, 13.3.2, 13.3.7
  • matrices as linear mappings: 11.2.3
  • additional exercises (see my email of 10/25): 11.2.6, 11.3.3, 11.4.3

Homework to turn in Mon. Oct. 29.

  • Problem 11-2 (p. 220).

Exercises on linear algebra to discuss Friday Nov. 2.

  • matrix rank: 12.4.3 (refers to 11.1.4)
  • determinant: 13.1.1
  • quadratic forms: 13.4.4; 13.4.5; 13.4.7 (a) and (b) (although they aren't lettered; so, do the first and second matrices)

Homework to turn in Mon. Nov. 5: TBA.

  • Exercise 13.1.2
  • Exercise 13.4.7(c) (that is, the third of the three matrices)

Supplemental materials, should you wish to consult them:

  • A nice 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, the following accessible supplementary treatment can be found FREE online. We are covering material corresponding to some of that in Chapters 1, 2, 4-12, and 16.
    • David Cherney, Tom Denton, Rohit Thomas, and Andrew Watson, Linear Algebra, online (1.4 megabytes, 460 pages), University of California, Davis (2013).
  • 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 his website from MIT.
  • Another alternative, and interesting, definition of the determinant, in terms of permutations of the columns is given here.


3. Multivariate calculus

One week, Mon. Nov. 5 through Fri. Nov. 9

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 beginning from top of p. 276
  • optimization: Read sections 16.1, 16.2
  • if time: implicit functions, section 15.1

Exercises on multivariate calculus to discuss Friday Nov. 9.

  • For practice on partial derivatives: 14.1.1(a through d), 14.2.2, 14.2.3
  • Critical points and second-order conditions: 16.1.1, 16.1.2 (a through d), 16.1.3 (a, b), 16.1.5 (a, b, c)
    • Hint for 16.1.5(c): consider movements along the line y = -x
  • Global optima, concavity, and convexity: 16.2.1 (for a, b, c only)

Homework to turn in Mon. Nov. 12.

  • 16.1.3 (c) and (d)


*** Second Exam out about Wed. Nov. 14 ***



4. Probability and Statistical Inference


  • Pemberton & Rao, excerpts from chapters 21 & 22
    • In section 22.2, we'll just cover the material on the variance, pp. 476-77.
  • Excerpts from  Wasserman, All of Statistics, chapters 1-6 and 10
    • We will cover the material in chapters 1-4 of Wasserman primarily by using chapters 21 & 22 in P&R.
    • We will use Wasserman chapter 5 instead of P&R section 22.4 to cover the Central Limit Theorem and related background.
    • We will introduce inference and hypothesis testing using sections 6.3, 10.1, 10.2, and 13.1 in Wasserman.

Probability and random variables

Nov 19 & 26

Chapter 21 in P & R (corresponds to chapters 1 and 2 in Wasserman)

Exercises to discuss Friday Nov. 30

  • In Section 21.1: Exercises 1, 3, 4, 5
  • In Section 21.2: Exercises 1, 3, 4
  • In Section 21.3: Exercises 2, 3, 5
  • In Section 21.4: Exercises 3


Expectations and Moments

Nov. 26

Chapter 22.1-22.3 (in section 2, through p. 478 only) in P & R (corresponds to chapter 3 in Wasserman)

Exercises to discuss Friday Nov. 30

  • In Section 22.1: Exercises 1, 2, 3
  • In Section 22.2: Exercises 1, 3, 8 (don't worry about the MGF in 8).

Homework to turn in Mon. Dec. 3

  • 21.4.2, 22.2.5

From Wikipedia article on "Correlation and Dependence": neat illustrations in which dependent variables have zero correlation


Convergence of Random Variables

Nov. 28

Wasserman, sections 5.1-5.4 (corresponds to section 22.4 in P&R)

  • convergence of random variables
  • Law of Large Numbers
  • Central Limit Theorem

Exercises to discuss Friday Dec. 7

  • P&R, Section 22.3: Exercises 1, 2, 4
  • Wasserman Chapter 3, Exercise 4
  • Wasserman 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.


Statistical Inference

Dec. 3

Wasserman, section 6.3

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


Hypothesis testing

Dec. 5

Wasserman, sections 10.1, 10.2; if time we'll touch on 13.1

  • 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 be assigned (subject to revision)

  • 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.)
    • Then 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.)



Third Exam

  • review session Mon. Dec. 11
  • Exam distributed by Tue. Dec. 11
  • Exam due date Sun. Dec. 16




This page written by Randall Calvert ©2018

Randy Calvert
Wash U email: calvert
office hours: Tues & Thur when you catch me in, or by appointment

Class meets Monday & Wednesday 10:00-11:30 in Seigle 303


Weekly Problem Session: Fridays 10-11:30 in Seigle 205

*** Note change in projected distribution date for Exam 2: Nov. 14 instead of Oct. 31

Jump directly to CURRENT topics & assignments


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.

  • Your grades for the course will be derived as follows. (Are you clear on the meaning of grades for a PhD student?)
  • Exams 75%.  Three, non-cumulative, open-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. The first two exams will be handed out on a Wednesday following a review session, and due back at the beginning of class the following Monday My preliminary schedule is as follows:
    • First exam out Wed. Sep. 26
    • Second exam out Wed. Nov. 14
    • Third exam out at the beginning of finals week (Thu. Dec. 13)
  • Homework 25%. Problems, drawn mostly from the textbooks, will be assigned as we go along on Mondays and Wednesdays, and always noted at the appropriate place on the Course Outline online.
    • Each Friday at 10-11:30 I will offer a problem help session. You should try all the week's problems by then. Your main source of official feedback on the problems will be my answers to your questions posed in the problem session. Attendance is optional, but I urge you to take advantage of these sessions.
    • Each week I will assign one or a few of the problems as ``Homework'' to be solved, typeset in LaTeX, and turned in (on paper or by email) at the beginning of class Monday. Homework problems will not be explicitly solved in the Friday help session, although we'll definitely work through similar problems.
    • Although I encourage you to work with other students on the weekly problems in general, I strongly suggest you do the Homework problems alone after you have jointly worked on problems, and even discussed the Homework problems themselves. This will let me to catch rough spots in your skills (as well as giving us needed LaTeX practice). However, it will not be possible to cover all techniques via Homework. The more general problems will be your only practice on some issues.
    • I will grade the Homework problem or problems each week, based heavily on completion and 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.