McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Problems 1, 2, 3, and 5 by Prof. Hundemer, problems 4 and 6 by Hang Lu Su.
The assignment was marked by Eric Hanson.
Assignment 3: Solutions
1. Let x be a real numbe
MATH 254 Hon. Analysis I
Prof. V. Jaksic
Solutions for Assignment II.
The solutions were written and the assignment graded by Renaud Raquepas.
1. Let A R, B R be two sets bounded from above. The sum of A and B is the set
A + B = cfw_a + b : a A, b B,
Prov
McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2014
Assignment 1
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1,2.
This assignment is due Tuesday, September 16, a
McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2014
Assignment 6
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1 and 2.
This assignment is due Tuesday, October 21,
McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2014
Assignment 7
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1 and 2.
This assignment is due Tuesday, October 28,
McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2014
Assignment 2
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1,2.
This assignment is due Tuesday, September 23, a
MATH 557 - ASSIGNMENT 2
To be handed in not later than 5pm, 14th February 2008.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1 Suppose that X1 , . . . , Xn are a random sample from the pdf
fX| (x|) =
1
expcfw_
MATH 557 - EXERCISES 2 These exercises are not for assessment
1 Suppose that X Binomial(n, ) for 0 < < 1. (a) Verify that the estimator T (X) = X/n is unbiased for . (b) Consider () = 1/. Find an unbiased estimator of (). 2 Suppose that X1 , . . . , Xn U
MATH 557 - MID-TERM 2008 - SOLUTIONS
1.
(a) Note rst that by standard expansion into a quartic polynomial
x
k
4
k
wj (, )xj = w0 (, ) +
= w0 (, ) +
j=1
wj (, )tj (x)
j=1
say, where wj (, ) are constant functions of and . Thus
k
fX|, (x|, ) = h(x)c(, ) ex
M ATH 557 - ASSIGNMENT 3
S OLUTIONS
1
(a) To nd the UMP test, consider
H0 : = 1
H1 : = 1
for 1 > 1. By Neyman-Pearson, the rejection region is constructed by looking at
n
fX | (x|1 )
fX | (x|1)
where T (x) =
n
1 (1 xi )1 1
=
i=1
= n cfw_T (x)1 1
1
1
(1 xi
M ATH 557 - ASSIGNMENT 3
To be handed in not later than 5pm, 20th March 2008.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1 Let X1 , . . . , Xn be a random sample from the Beta(1, ) probability model, for par
CHAPTER I. SHEAVES AND ALGEBRAIC TOPOLOGY
Sheaf theory was originally a part of algebraic topology; namely, that part
concerned with studying the various kinds of chain complexes, like the Alexander
complex or the De Rham complex, which involve chains wit
CHAPTER II. SHEAVES AND COMPLEX ANALYSIS
The definition in Leray [2] 1946 of a normal (or continuous) faisceau I immediately suggests looking at the module B assigned to a (closed) point x ~ X .
x
Continuity implies that it is the (direct) limit of the mo
MA 533, Partial Dierential Equations, Fall 2013
The primary goal of this course is to introduce some of basic methods in the
theory of linear partial dierential equations.
Time and Place: 11:00am11:50am, MWF, CB 347.
Instructor: Changyou Wang, Professor o
MATH 533 Final Examination December 9th, 2008
Student Name:
Student Number:
McGill University
Faculty of Science
FINAL EXAMINATION
MATH 533
Regression and Analysis of Variance
December 9th, 2008
9 a.m. - 12 Noon
Calculators are allowed.
One 8.5 11 two-sid
MATH 556 - PRACTICE EXAM QUESTIONS
1. The joint pdf for continuous random variables X, Y with ranges X Y R+ is given by
1
fX,Y (x, y) = c1 exp (x + y)
2
x, y > 0
and zero otherwise, for some normalizing constant c1 .
Consider continuous random variable U
MATH 556 - ASSIGNMENT 1 SOLUTIONS
1. For the discrete variables concerned
(a) As
x=0 y=0
(x + y)y
x
y
y
=
=
x
+
x!
y!
x!
y!
(y 1)!
x=0
y=0
x=0
y=0
y=1
x
y
x
y
=
x
=
xe + e
+
x!
y!
y!
x!
x+y
(x + y)
x!y!
x
x=0
y=0
= e
x=1
x=0
y=0
x
+
(x 1)!
= e (
MATH 556 - ASSIGNMENT 1
To be handed in not later than 5pm, 28th September 2006.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1. Suppose X and Y are discrete random variables having joint pmf given by
fX,Y (x,
MATH 556 - ASSIGNMENT 3
To be handed in not later than 5pm, 16th November 2006.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1 Consider the three-level hierarchical model:
LEVEL 3 : > 0, r cfw_1, 2, . . .
Fixe
MATH 556 - ASSIGNMENT 2
To be handed in not later than 5pm, 19th October 2006.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1
(a) Suppose that X is a continuous rv with pdf fX and characteristic function (cf)
MATH 556 - ASSIGNMENT 3 SOLUTIONS
1
(a) By iterated expectation, using the formula sheet to quote expectations for Gamma and Poisson
Ef [N ] + r/2
N + r/2
+ r/2
= N
=
= 2 + r
1/2
1/2
1/2
EfX [X] = EfN [EfX|N [X|N = n] = EfN
3 M ARKS
(b) By the same metho
MATH 556 - ASSIGNMENT 4 SOLUTIONS
1 For t > 0, and constant k > 0
P [X t] = P [X + k t + k] P [(X + k)2 (t + k)2 ]
EfX [(X + k)2 ]
(t + k)2
by the Chebychev Lemma. Now if k = 2 /t, then
P [X t]
EfX [(X + 2 /t)2 ]
Ef [(tX + 2 )2 ]
= X 2
(t + 2 /t)2
(t +
MATH 556 - MID-TERM SOLUTIONS
1.
(a) (i) From rst principles (univariate transformation theorem also acceptable): for 0 < x < 1
2
2
arcsin x = arcsin x
FX (x) = P [X x] = P [sin(U/2) x] = P U
and zero otherwise, as the sine function is monotonic increasi
MATH 556 - PRACTICE EXAM QUESTIONS II SOLUTIONS
1.
(a) Using properties of cfs, we have
CTn (t) = e|t|
n
= e|nt|
Now using the scale transformation result for mgfs/cfs (given on Formula Sheet), we have
that if V = U , then
CV (t) = CU (t)
we deduce that,
MATH 556 - MID-TERM EXAMINATION
Marks can be obtained by answering all questions. All questions carry equal marks.
1.
(a) Suppose that U is a continuous random variable, and U U nif orm(0, 1). Let random
variable X be dened in terms of U by
X = sin(U/2).
MATH 556 - PRACTICE EXAM QUESTIONS
1. Due to the symmetry of form, this joint pdf factorizes simply as
fX,Y (x, y) =
x
c1 exp
2
y
c1 exp
2
= fX (x)fY (y)
x, y > 0
and hence the variables are independent. Now
exp
0
x
dx = 2
2
so therefore c1 = 1 , and h
MATH 556 - ASSIGNMENT 2 SOLUTIONS
1
(a) (i) By direct calculation
CX (t) = EfX [eitX ] =
eitx expcfw_x ex dx =
expcfw_(1 it)x ex dx
=
y it ey dx = (1 it)
0
after setting y = ex .
Note that the Gamma function notation usage here is legitimate; the Gamma
MATH 556 - ASSIGNMENT 4
To be handed in not later than 5pm, 30th November 2006.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1 Suppose that X has expectation zero, and nite variance 2 . Prove that, for t > 0,
MATH 556 - PRACTICE EXAM QUESTIONS II
1. Suppose that X1 , X2 , . are i.i.d Cauchy random variables with pdf
fX (x) =
1 1
1 + x2
xR
and characteristic function CX (t) = expcfw_|t|.
(a) Find the distribution of the random variable Tn dened by
n
Tn =
Xi .