Math 3220-1 Final, December 17, 2015
Solutions
Problem 1. Let x1 , x2 , . . . , xn be a family of mutually orthogonal vectors in Rd . Show that
x1 + x2 + + xn
2
= x1
2
+ x2
2
+ + xn 2 .
Solution: We p
Math 3220-1 Midterm 2, November 18, 2015
Solutions
Problem 1 (20 points). Let f : R R be a dierentiable function
on R. Dene a function F : R2 R by F (x, y) = f (x + y). Show
that
F
F
=
.
x
y
Solution:
Math 3220-1 Midterm 1, September 30, 2015
Solutions
Problem 1 (20 points). Show that
1
xy =
x+y
4
for any x, y Rd .
2
xy
2
Solution: We have
1
1
( x + y 2 x y 2 ) = (x + y) (x + y) (x y) (x y)
4
4
1
MATH 5040/6810: Homework 5 (Due Monday, December 7)
Problem 1. Assume that Archie, Betty and Veronica are shopping in Smiths for the
Thanksgiving dinner. It is so late in the day that they are the onl
MATH 5040/6810: Homework 4 (Due Monday, November 23)
Problem 1. Consider the following branching process Xn . Generation 0 has 1 individual,
i.e. X0 = 1. Then, each individual of each generation, inde
MATH 5040/6810: Homework 3 (Due Monday, November 2)
Solutions are in blue ink. Red ink is for alternate solutions.
Problem 1
Prove that simple symmetric random walk is recurrent in two dimensions. (Hi
MATH 5040/6810: Homework 2 (Due Monday, September 28)
Problem 1
Let N 4 be an integer. Consider a Markov chain with state space cfw_1, 2, 3, . . . , N .
From j N 3 the chain moves to j + 2 or j + 3 eq
MATH 5040/6810: Homework 1 (Due Fri, Sep. 11, at the beginning of lecture)
Problem 1
a) Let A0 , A1 , A2 , A3 be events of non-zero probability. Show that
Pcfw_A0 A1 A2 A3 = Pcfw_A0 Pcfw_A1 |A0 Pcfw_
Misty Rasmussen
Homework 4
1.14: Let X_n be a Markov Chain on state space cfw_1,2,3,4,5 with transition matrix:
a) Is the chain irreducible? Is it aperiodic?
P=
1
2
3
4
5
1
0
0.5
0.5
0
0
2
0
0
0
0.2
0
Quiz 4 (Feb 18, 2015), Math 5010-002 Name:
NO WORK means NO POINTS!
1. (10 pt) An urn contains 5 balls numbered from 1 to 5. We draw 3 of them at random
without replacement.
(a) Let X be the largest n
Exam 1
Math 5010 Section 2
February 23, 2015
NAME:
(Please Print)
DIRECTIONS:
Do NOT open the exam until you are told to do so.
Do each of the problems and show all your work.
NO WORK means NO POIN
Exam 2
Math 5010 Section 2
April 6, 2015
NAME:
(Please Print)
DIRECTIONS:
Do NOT open the exam until you are told to do so.
Do each of the problems and show all your work.
NO WORK means NO POINTS!
Math 5080-2, F 2015. Assignment 2. Chapter 6, Exercises 18, 25, 27, 28, 29,
31.
18. (a) X = T and Y = S T , so
0
J = det
1
1
1
= 1,
and fS,T (s, t) =Rf (t, s t) = e(st) , where 0 < t < s t < , or s >
Quiz 5 (March 2, 2015), Math 5010-002
Name:
NO WORK means NO POINTS!
1.
(10 pt) Let X be a random variable with probability density function given by
o
ZI
(a)FindPcfw_X>.
r4
(b) Find Pcfw_X
2
2-
2X <0
Dan Shi
3070 R Lab Assignment 3
Due in class on THURSDAY, January 24
ANNOTATE ALL OF YOUR CODE
The data sets 'normtemp1.csv' and 'normtemp2.csv' contain normal body
temperature
measures for 130 health
Dan Shi
Math 3010
Keplers Law
Kepler was a German mathematician in the 17th century. He is well known for his
laws of planetary motion. This work is very important for later scientific discoveries. Fo