Math (P)refresher for Political Scientists
Harvard University
2011
The documents in this booklet are the product of generations of Math (P)refresher Instructors: Curt Signorino
1996-1997; Ken Scheve 1997-1998; Eric Dickson 1998-2000; Orit Kedar 1999; Jame
August 2010, Marc-Andr Letendre
e
Review of Mathematics for Economics:
Probability Theory
Linear and Matrix Algebra
Dierence Equations
Marc-Andr Letendre
e
McMaster University
Page i
August 2010, Marc-Andr Letendre
e
Page ii
Contents
1 Matrix Algebra
1
Problems for Section 3
MA Methods Review
Jerey Penney
Department of Economics, Queens University
#1 Modify the following expression to only use a single radical sign:
5
4
3
2
2
#2 Using summation notation, reduce the following sums as much as possible:
(a
Problems for Section 5
MA Methods Review
Jerey Penney
Department of Economics, Queens University
#1 Using the denition of a derivative, nd the derivative of 5x2 + 3x 6.
#2 Solve x [f (x, y)] for f (x, y) = x2 y 3 ; y = y(x) = 3x + 2x2 + 4. Only substitute
Problems for Section 2
MA Methods Review
Jerey Penney
Department of Economics, Queens University
#1 Find simpler expressions of these formulas:
(a) (P Q)
(b) (Q P ) P
(c) P (Q P )
(d) (P (Q R) Q
#2 Suppose A = cfw_1, 2, 3, 4, 5 and B = cfw_2, 4, 6, 8, 10.
Problems for Section 4
MA Methods Review
Jerey Penney
Department of Economics, Queens University
#1 Let x be a column vector. Show that norm of the following expression is equal to 1
when using the Euclidean norm. Hint: its probably easier that you stick
Problems for Section 7
MA Methods Review
Jerey Penney
Department of Economics, Queens University
#1 Let y1 = 5x1 + 3x2 and y2 = 25x2 + 30x1 x2 + 9x2 . Are y1 and y2 functionally
1
2
dependent?
#2 Find the Hessian for f (l, k; A) = Al k , where + < 1, and
Problems for Section 8
MA Methods Review
Jerey Penney
Department of Economics, Queens University
#1 Consider the expression dy + ay = 0, where y = y(t). Use integration to reach the
dt
result y(t) = Qeat where Q is a constant.
#2 Find the solution to the
Problems for Section 6
MA Methods Review
Jerey Penney
Department of Economics, Queens University
#1 Show that the innite sum p + 2p(1 p) + 3p(1 p)2 + . . . is equal to p1 , where
0 < p < 1. This is the same problem as in Section 3 problem #2 (b), except y
Queen's University
Department of Economics
Fall 2014
Graduate Methods Review
Monday to Friday 9:00 12:00 @ Dunning 213
instructor: Mr. Jeffrey Penney
email: [email protected]
office: Dunning 347
office hours: immediately after class or by appointmen