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_A2 |A0 A1 Pcfw_A3 |A2 A1 A0
Solution:
Pcfw_A0 Pcfw_A1
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 equally likely. From N 2 the chain
moves to N or 1 equall
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 prove the result by induction in n. If n = 1 the
stateme
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. (Hint: adapt
the computations we did in class for the one-
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, independently of all other individuals,
gives at most 2 des
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 only customers in the store.
There are two cashiers at the
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
= (x x + 2x y + y y x x + 2x y y y) = x y.
4
Problem 2
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: By the chain rule have
F
F
= f (x + y) and
= f (x + y)