Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
AMS 205  Homework
Chapter 8  Hypothesis Testing
Solutions
8.5 a. The loglikelihood is
log L(, x) = n log + n log ( + 1) log(
xi ),
x(1) ,
i
where x(1) = mini xi . For any value of , this is an increasing function of for x(1) . So both
the restricted
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
AMS 205  Homework
Chapter 2  Transformations and Expectations Solutions
2.1 a. fx (x) = 42x5 (1 x), 0 < x < 1; y = x3 = g(x), monotone, and = (0, 1). Use Theorem 2.1.5. fY (y) = fx (g 1 (y) d 1 d 1 g (y) = fx (y 1/3 ) (y 1/3 ) = 42y 5/3 (1 y 1/3 )( y 2/
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Math 354 Fall 09, Problem set 2
Due Wed. Oct. 21
Hand in only the starred problems.
In this problem set use only results from Chapt. 1 and 2, for example the
residue theorem is forbidden.
Stein Chapt. 1 :
18*. Justify any rearrangement of series that you
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
MAS541 COMPLEX FUNCTION THEORY
FALL 2010  MIDTERM EXAM  22 OCTOBER
No document or electronic device allowed. Papers written neatly receive up
to 10 points (out of 100 points). Good luck!
(1) 8 pts Let f O(C). Assume a, b R>0 and k Z>0 such that
R > 0,
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
MAS541 COMPLEX FUNCTION THEORY
FALL 2009  MIDTERM SOLUTIONS
(1) (a) The picture is a centered annulus with radii e and e .
(b) For all z C, ef (z ) eM , so by Liouvilles theorem, exp f is
constant. As the bers of exp are discrete and C is connected,
the
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
MAS541 COMPLEX FUNCTION THEORY
FALL 2009  MIDTERM EXAM  23 OCTOBER
No document or calculator allowed. Papers written neatly receive up to 10
points (out of 100 points). Good luck!
(1) 18 pts(a) Let and be real numbers such that < , and let
S = cfw_z C
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Math 302: Solutions to Homework
Steven Miller
December 14, 2010
Abstract
Below are detailed solutions to the homework problems from Math 302
Complex Analysis (Williams College, Fall 2010, Professor Steven J. Miller,
sjm1@williams.edu). The course homepage
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
PROBLEM SET NUMBER 1
MAT354 FALL 2010
Solution to Problem: 1 (6 page 26). If a Cz take B to be an open disc,
small enough to be contained in . This is possible since is open. Since any
two points in B can be joined by a segment then z can be joined to any
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Math 113 (Spring 2009) YumTong Siu
1
Solution of Homework Assigned on February 5, 2009
due February 17, 2009
Problem 1 (from Stein & Shakarchi, pp.3031, #25).
(a) Evaluate the integral
z n dz
for all integers n (positive, negative, or zero). Here is any
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Problems and Solutions in
R EAL AND C OMPLEX A NALYSIS
William J. DeMeo
July 9, 2010
c William J. DeMeo. All rights reserved. This document may be copied for personal use. Permission to reproduce
this document for other purposes may be obtained by emailin
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Homework 1 STAT 543
On campus: Due Friday, January 19 by 5:00 pm (TAs ofce);
you also may turn in the assignment in class on the same Friday
Distance students: Due Wednesday, January 24 by 12:00 pm (TAs email)
1
1. For X1 , . . . , Xn , show that 2 (1 )2
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Homework 12 STAT 543
Not to turn in  practice only
1. Let X1 , . . . , Xn be iid random variables with the common cdf
F (x, ) =
0
if x
3
1 (x/)
if x ,
where > 0.
(a) Find the cdf of Tn X(1) / where X(1) denotes the minimum of X1 , . . . , Xn .
(b) Find
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Homework 11 STAT 543
On campus: Due Friday, April 20 by 5:00 pm (TAs ofce);
you also may turn in the assignment in class on the same Friday
Distance students: Due Friday, April 27 by 5:00 pm (TAs email)
1. Let X1 , . . . , Xn be iid exponential() and let
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Homework 10 STAT 543
On campus: Due Friday, April 13 by 5:00 pm (TAs ofce);
you also may turn in the assignment in class on the same Friday
Distance students: Due Friday, April 20 by 5:00 pm (TAs email)
1. Problem 9.4, Casella and Berger (2nd Edition)
2.
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Homework 9 STAT 543
Due Friday, March 30 by 5:00 pm (TAs ofce);
you also may turn in the assignment in class on the same Friday
Distance students: Due Friday, April 6 by 5:00 pm (TAs email)
1. Consider one observation X from the probability density functi
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Homework 8 STAT 543
Due Friday, March 23 by 5:00 pm (TAs ofce);
you also may turn in the assignment in class on the same Friday
Distance students: Due Friday, March 30 by 5:00 pm (TAs email)
1. Problem 8.19, Casella and Berger (2nd Edition)
2. Problem 8.2
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Homework 7 STAT 543
Due Friday, March 9 by 5:00 pm (TAs ofce); you also may turn in the assignment in class on the same Friday Distance students: Due Friday, March 16 by 5:00 pm (TAs email) 1. (a) Problem 8.13(a)(c), Casella and Berger (2nd Edition) (b)
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Math 354 Fall 2010, Problem set 1
Due Fri. Oct. 8
Hand in only the starred problems. Note that there are two pages in
this document.
About starred and unstarred problems: The starred problems are
a (not entirely) random selection from the whole set. The r
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
AMS 205  Homework
Chapter 7  Point Estimation
Solutions
7.1 For each value of x, the MLE is the value de that maximizes f (x). These values are in the
following table.
x
0
1
2
3
4
1
1
2 or 3
3
3
At x = 2, f (x2) = f (x3) = 1/4 are both maxima, so bot
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
AMS 205  Homework
Chapter 6  Principles of Data Reduction
Solutions
6.1 By the Factorization Theorem, X  is sucient because the pdf of X is
1
1
2
2
2
2
f (x 2 ) =
ex /2 =
ex /2 = g (x 2 ). 1 .
2
2
h(x)
6.2 By the Factorization Theorem, T (X ) =
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
AMS 205  Homework
Chapter 5  Properties of a Random Sample
Solutions
5.3 Note que Yi Bernoulli with pi = P (Xi ) = 1 F () for each i. Since the Yi s are iid
Bernoulli,
n
i=1 Yi
binomial (n, p = 1 F ().
5.10 a.
1 = EXi =
2 = E (Xi )2 = 2
3 = E (Xi )3
=
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
AMS 205  Homework
Chapter 4  Multiple Random Variables Solutions
4.1 Since the ditribution is uniform, the easiest way to calculate these probabilities is as the ratio of areas, the total area being 4. a. The circle x2 + y 2 1 has area , so P (X 2 + Y 2
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
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Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
AMS 205  Homework
Chapter 2  Transformations and Expectations Solutions
2.1 a. fx (x) = 42x5 (1 x), 0 < x < 1; y = x3 = g(x), monotone, and = (0, 1). Use Theorem 2.1.5. fY (y) = fx (g 1 (y) d 1 d 1 g (y) = fx (y 1/3 ) (y 1/3 ) = 42y 5/3 (1 y 1/3 )( y 2/
Korea Advanced Institute of Science and Technology
Statistical Inference
MATH MAS555

Fall 2011
Problems and Solutions in
R EAL AND C OMPLEX A NALYSIS
William J. DeMeo
July 9, 2010
c William J. DeMeo. All rights reserved. This document may be copied for personal use. Permission to reproduce
this document for other purposes may be obtained by emailin