APMA 1650/1655 Homework 2
February 15, 2016
Due before class on Friday, Feb. 19th. It can be dropped off in the APMA 1650 homework box on
the first floor of the APMA department, 182 George St OR at class (before it starts) on Friday.
APMA 1650: Complete a
Homework 6
1. (a) To find the marginal density for a random variable we need to integrate over the other random
variable(s).
Z
f (x, y)dy
fx (x) =
Z
e(x+y) dy
=
0
= ex
when x > 0 (and 0 otherwise)
Z
f (x, y)dx
fy (y) =
Z
=
e(x+y) dx
0
= ey
when y > 0 (a
APMA 1650/1655 Homework 8
April 16, 2016
Due before class on Friday, April 22nd. It can be dropped off in the APMA 1650 homework box
on the first floor of the APMA department, 182 George St OR at class (before it starts) on Friday.
APMA 1650: Complete all
5. Excited about your scheme to estimate , you line up in front of your poster, darts in hand, when suddenly it
dawns on you that you cant actually throw darts uniformly they always seem to go to the right. Precisely,
the distribution of your dart throws
APMA 1650/1655 Homework 4
March 4, 2016
Due before class on Friday, March 11th. It can be dropped off in the APMA 1650 homework box
on the first floor of the APMA department, 182 George St OR at class (before it starts) on Friday.
APMA 1650: Complete all
APMA 1650/1655 Homework 3
February 19, 2016
Due before class on Friday, Feb. 26th. It can be dropped off in the APMA 1650 homework box on
the first floor of the APMA department, 182 George St OR at class (before it starts) on Friday.
APMA 1650: Complete a
APMA 1650/1655 Homework 9
April 22, 2016
Due before class on Friday, April 22nd. It can be dropped off in the APMA 1650 homework box
on the first floor of the APMA department, 182 George St OR at class (before it starts) on Friday.
APMA 1650: Complete all
APMA 1650/55: Problem Set 2
February 17, 2016
Problem 1
(a)
We seek P(Player 1 wins).
Note: A round is a single coin toss. So player 1 tosses in round 1, player 2 in round 2, and so on. Thus, player 1
wins on H, TTH, TTTTH, .
P (H) = 12
2
P (T T H) = 12 1
APMA 1650/1655 Homework 6
March 18, 2016
Due before class on Friday, March 24th. It can be dropped off in the APMA 1650 homework box
on the first floor of the APMA department, 182 George St OR at class (before it starts) on Friday.
APMA 1650: Complete all
APMA 1650/1655 Homework 7
April 9, 2016
Due before class on Friday, April 15th. It can be dropped off in the APMA 1650 homework box on
the first floor of the APMA department, 182 George St OR at class (before it starts) on Friday.
APMA 1650: Complete all
APMA 1650/1655 Homework 1
February 1, 2016
Due before class on Friday, Feb. 12th. It can be dropped off in the APMA 1650 homework box on
the first floor of the APMA department, 182 George St OR at class (before it starts) on Friday.
APMA 1650: Complete al
APMA 1650/55 HOMEWORK 1 SOLUTIONS
1. Suppose you flip a fair coin 20 times.
(a) What is the probability of flipping 4 Heads? 8 Heads? 10 Heads?
Solution: For exactly 5 heads, we have 20 flips, and we want to pick the 4 of them
that will be heads. This can
Homework 9
1. (a) There is only one parameter, , so we only need one moment:
1 = E[Y ] =
1
so we can get the method of moments estimator by solving
n
1
M M
yielding
1X
=
Yi
n i=1
n
M M = Pn
i=1
(b) The likelihood is
f=
n
Y
Yi
Pn
f (yi |) = n (1 )
i=1
Yi n
MATH 825 FALL 2014
ANALYSIS AND GEOMETRY IN CARNOT-CARATHEODORY SPACES
Contents
1. Introduction
1.1. Vector elds and ows
1.2. Metrics
1.3. Commutators
1.4. Consequences of Hrmanders condition
o
1.5. Sums of Squares and Variants
1.6. Four Classical Example
Functiones et Approximatio
XXXIX.2 (2008), 191204
SUMMATION METHODS AND DISTRIBUTION OF
EIGENVALUES OF HECKE OPERATORS
Sanoli Gun[1] , M. Ram Murty[2] , Purusottam Rath[3]
Dedicated to Professor Wadysaw Narkiewicz
Abstract: Let p be a xed prime number. Le
Practice Midterm
1. Suppose that f is continuous and 2-periodic. Is it true that
2
|SN f (x) f (x)|dx = 0?
lim
N
0
Explain your answer.
2. Let gN be the 2-periodic function dened on [, ] by setting gN (x) = N
if |x| 1/N and gN (x) = 1/N if 1/N |x| . Supp
WRITTEN HOMEWORK #5, DUE 4/30/2012 AT 4PM
You can turn this in during class on April 30 or at my office (Kemeny 316) by 4pm.
Please make sure your homework assignment is stapled, if necessary, before handing
it in. (In particular, there is no guarantee th
Math 403 Lecture Notes for Week 2
1
Limsup and Liminf
Definition 1. Let A R. If A has an upper bound, we let sup A be the least
upper bound of A. When A is unbounded we sometimes write sup A = .
If A has a lower bound, we let inf A be the greatest lower b
Introduction to Fourier Analysis
Home assignment 4
1. Let f : [, ] R, f () = . Use Parsevals formula to compute the
sums
X
X
1
1
and
.
4
n4
(2n + 1)
n=0
n=1
Solution. In the fifth exercise of the first exercise set we proved that the
Fourier coefficient
Introduction to Fourier Analysis
Home assignment 5
1. Assume that f is a 2-periodic integrable function. Show that for all
n Z \ 0 , we have
1
f (n) =
2
f x+
einx dx,
n
and hence
1
f (n) =
4
f (x)f x +
n
einx dx.
Solution. Because of the 2-periodicity of
Introduction to Fourier Analysis
Home assignment 6
1. Let : [a, b] R2 be a dierentiable parametrization for the closed
curve . Prove that it is a parametrization by the arc length if and only if
the length of the curve from (a) to (s) for all s is equal t
Introduction to Fourier Analysis
Home assignment 7
1. Let (x) = x for x [0, 1]. Extend to R as a 2-periodic function. Prove
that the formula
n
X
3
f (x) =
(4n x)
4
n=0
defines a continuous function on R.
Solution. Since (x) 6 1 for all x R, the series
Equidistribution and Weyls criterion
by Brad Hannigan-Daley
We introduce the idea of a sequence of numbers being equidistributed (mod 1), and we state and prove a
theorem of Hermann Weyl which characterizes such sequences. We also discuss a few interestin
Introduction to Fourieranalysis
Home assignment 6
1. Let : [a, b] R2 be dierentiable parametrization for the closed curve
. Prove that it is a parametrization by the arclength if and only if
the length of the curve from (a) to (s for all s is equal to s a
Research Journal of Finance and Accounting
ISSN 2222-1697 (Paper) ISSN 2222-2847 (Online)
Vol.5, No.7, 2014
www.iiste.org
Analysis of Performance and Financial soundness of financial
institution (Banks): A Comparative Study
Md. Qamruzzaman
CMA ICMAB, MBA