Problem Set 3: The Short, the Long and Factors
on the Run
ECON 3004: International Trade
University College London
Due by 12.00 pm on the 15th of November, 2013
Important notice: Problem sets must be submitted through Turnit-in (Moodle, ECON3004, Topic 3)
Math 2101
Homework 9
NOT DUE
While this assignment will not be marked, please study Part A for the nal exam in
May.
PART A
1. (a) How many roots of the equation z 4 6z + 3 = 0 have their modulus between 1
and 2?
Let f (z ) = z 4 and g (z ) = 6z + 3. On th
Math 2101
Homework 2
Due: October 17, 2012
1. Verify the Cauchy-Riemann equations for f (z ) = z 3 and f (z ) = ex (cos y + i sin y ).
For the function f (z ) = z 3 we have u = x3 3xy 2 and v = 3x2 y y 3 and
u
v
= 3x2 3y 2 =
,
x
y
u
v
= 6xy = .
y
x
For th
Math 2101
Homework 1: Solutions
Due: October 10, 2012
The homework consists mostly of a selection of problems from the suggested books.
1. (a) Find the value of (1 + i)n + (1 i)n for every n N.
We will use the polar form of 1 + i = 2(cos(/4) + i sin(/4) a
Math 2101
Homework 1
Due: October 10, 2012
Problems marked with will be marked. The students are advised to work over ALL
problems in the assignments, so as to keep pace with the material.
1. (a) Find the value of (1 + i)n + (1 i)n for every n N.
(b) Show
Math 2101
Homework 4
Due: October 31, 2012
1. (a) Solve the equation ez = 1 + 3 i, i.e. nd all complex numbers z satisfying it.
We write 1 + 3i in polar form. Since |1 + i 3| = 2 and the argument can be taken
to be /3 = arctan 3, we have 1 + i 3 = 2ei/3 .
Math 2101
Homework 4
Due: October 31, 2012
1. (a) Solve the equation ez = 1 + 3 i, i.e. nd all complex numbers z satisfying it.
(b) Show that ez = ez .
(c) Find the image of the semi-innite strip x 0 and 0 y under the transformation w = ez . Exhibit corre
Math 2101
Homework 5
Due: November 14, 2012
1. In the following you are not allowed to use the residue theorem or Cauchys integral
formula, as they have not been discussed yet.
(a) Compute the integral
x dz
|z |= r
for the positive sense of the circle, in
Math 2101
Homework 5
Due: November 14, 2012
1. In the following you are not allowed to use the residue theorem or Cauchys integral
formula, as they have not been discussed yet.
(a) Compute the integral
x dz
|z |= r
for the positive sense of the circle, in
Math 2101
Homework 7
Due: November 28, 2012
1. Let f be entire.
(a) Show that, if ef is bounded, then f is constant.
(b) Show that, if
(f ) is bounded above, then f is constant.
2. Let a, b R. Let f be entire and satisfy
f (z + a) = f (z ),
f (z + ib) = f
Math 2101
Homework 7
Due: November 28, 2012
1. Let f be entire.
(a) Show that, if ef is bounded, then f is constant.
Since ef is entire, as composition of ew and f , and bounded, by assumption, we can
apply Liouvilles theorem and deduce that ef is constan
Math 2101
Homework 6
Due: November 21, 2012
1. Let C be the circle |z i| = 2 traversed anticlockwise. Compute
C
cos z
dz,
z (z 2 + 8)
C
cosh z
dz,
z4
C
(z 2
1
dz.
+ 4)2
Do not use the residue theorem.
2. By evaluating the integral
1
2i
C
1
dz
(z a)(z a1 )
Math 2101
Homework 6
Due: November 21, 2012
1. Let C be the circle |z i| = 2 traversed anticlockwise. Compute
C
cos z
dz,
z (z 2 + 8)
C
cosh z
dz,
z4
C
(z 2
1
dz.
+ 4)2
Do not use the residue theorem.
(a) Using partial fractions we easily see that
1
z
1
=
M3508 Exercises 6
November 2013
1. Conditional expectation
Consider the following ltration (Pt )T=1 , where r = 0:
t
1
2
3
4
S (0, ) S (1, )
100
120
100
120
100
90
100
90
S (2, )
150
110
110
70
(i) Calculate the conditional probabilities for each edge of
MATH508 Exam solutions.
2010
1. (a) Write down the general principle used for valuing forwards. Using
this, calculate the one year future on Morgan Stanley (MS) shares given
that:
(i) MS is currently trading at 30 USD per share
(ii) USD interest rates are
MATH508 Exam solutions.
2011
1. (a) Write down the general principle used for valuing forwards.
Using this, calculate the 3-month future on Oil given the following information:
(i) Spot Oil is trading at 90USD per barrel.
(ii) USD interest rates are at 12
Math 2101
Homework 8
Due: December 5, 2012
1. Explain why the following integrals have the given value using the residue theorem.
Complete explanations are required.
/2
(a)
0
1
dt =
, a > 0, (b)
a + cos2 t
2 a2 + a
2
cos2 (3t)
3
(c)
dt =
.
5 4 cos(2t)
8
Math 2101
Homework 8
Due: December 5, 2012
1. Explain why the following integrals have the given value using the residue theorem.
Complete explanations are required.
/2
(a)
0
1
,
dt =
2t
a + cos
2 a2 + a
2
(c)
0
a > 0,
(b)
0
log x
dx = 0,
x2 + 1
2
cos (
YP: STAT2001/3101, 2013 2014
1
Combinatorics of the Multinomial Distribution
As quite a few students seemed to have problems with the combinatorics of the multinomial distribution, I thought a simple reference to the STAT1005 lecture notes (p.12)
might no
YP: STAT2001/3101, 2013 2014
1
The Covariance Matrix
Given a random vector X = (X1 , X2 , . . . , Xn )T , i.e. a collection of random variables
X1 , X2 , . . . , Xn , one can dene the covariance matrix of that vector:
cov(X1 , X1 ) cov(X1 , X2 ) . . . cov
YP: STAT2001/3101, 2013-2014
12
Exercise Sheet 4
The exercise sheet is subdivided into three parts:
Part A contains relatively straightforward questions which you should attempt
in your own time full solutions for these will be available on Moodle after
YP: STAT2001/3101, 2013-2014
3
Solutions to Exercise Sheet 2
A: Warm-up questions
1. (a) P (A) = P (cfw_1, 2) = P (cfw_1 cfw_2) = P (cfw_1) + P (cfw_2) = 1/2 + 1/4 = 3/4.
(b) P (X = 1|X A) = P (X = 1|A) =
P (X =1 and X A)
P (X A)
=
1/2
3/4
= 2/3
2. (a) It
YP: STAT2001/3101, 2013-2014
6
Exercise Sheet 2
The exercise sheet is subdivided into three parts:
Part A contains relatively straightforward questions which you should attempt
in your own time full solutions for these will be available on Moodle after t
Math 3109
Homework 5
Due: November 15, 2013
1. (a) Prove that, if f : [a, b] [c, d] R is continuous, then f is uniformly continuous,
i.e.
> 0 > 0 : |(x1 , x2 ) (y 1 , y 2 )| < = |f (x1 , x2 ) f (y 1 , y 2 )| < .
(b) Prove that, if f : [a, b] [c, d] R is
Math 3109
Homework 4
Due: November 1, 2013
1. (a) If f : R R satises f (a) = 0 for all a R, show that f is injective on all of R.
(b) (i) Dene f : R2 R2 by f (x, y ) = (ex cos y, ex sin y ). Show that detf (x, y ) = 0
for all (x, y ) but f is not injectiv
STAT2001/STAT3101:
Probability and Inference
Lecturer: Yvo Pokern,
Room 144 Dept. of Statistical Science
Lectures:
Christopher Ingold XLG1 on Thursdays 1pm-2pm
Anatomy G29 J. Z. Young Lecture Theatre on Fridays
2pm-4pm
Ofce Hour: Fridays 4pm-5pm
Email: