IE 5531 Final Exam - Dec 2013
Page 1 of 9
F INAL E XAM S OLUTION
IE 5531
Dec 11th, 2013
INSTRUCTIONS
a) Write ALL your answers in this exam paper.
b) Two pieces of notes are allowed. No computer or cell phone is allowed.
c) The exam time is 10:15am - 12:1

Problem 1. Let (Mn ) be a martingale with respect to (Xn ). Show that
E[Mm |Av ] = Mn ,
where
Av = cfw_Xn = xn , . . . , X0 = x0 , M0 = m0 .
Problem 2. Suppose (Mn ) is a martingale with respect to (Xn ) and is predictable. Show
that Mn = M0 .
Problem 3.

The problems from the book: 5.4, 5.5, 5.15 and the following problems. Denote by
cfw_B(t); t 0 a standard Brownian motion.
Problem 1. Let W (t) = B(a2 t)/a for a > 0. Verify that W (t) is also a standard Brownian
motion.
Problem 2. Compute the condition d

4.7
Solution: Note that if there is no customer in the store, then the arrival rate of the two
servers equals /2. If at least one of them is already busy, then the arrival rate is . On the
other hand, let ti be the service time of the i-th server, i.e., t

5.2 without the second half of (a)
Solution: (a) Let Av = cfw_Xn = xn , . . . , X0 = x0 , M0 = m0 . Then by the Markov property,
N
E[Xn+1 Xn |Av ] =
iP (Xn+1 = i|Av ) xn
i=0
N
ip(xn , i) xn
=
i=0
N
=
i
i=0
N
i
xn
N
i
1
xn
N
N i
xn
= xn xn = 0,
where we u

3.6
Solution: This is essentially an alternating renewal process with three states: state i means
that the child i shoots the basket with success probability pi until he or she fails. For j 1,
1 2 3
let rj , rj , rj be the numbers of times that the childr

1.54
Proof. Suppose that p is reversible with respect to some stationary distribution , i.e.,
i pij = j pji
for all i, j. Then
pij = j pji /i ,
which implies that
pij pjk pki =
j pji k pkj i pik
= pik pkj pji .
i j k
Suppose that p satises pij pjk pki = p

2.43
Solution: Let NF and NM be the Poisson processes with rate 30 and 20 per hour, respectively.
Since they are independent, Theorem 2.13 (superposition) implies that N (t) := NF (t)+NM (t)
is a Poisson process with rate 50 and for any new customer, ther

Problem 1. (Polya urn) An urn contains b black and r red balls. A ball is drawn at random.
It is replaced and moreover c balls of the color drawn are added. Let X0 = b/(b + r) and let
Xn be the portition of black balls attained at state n; that is, just a

1. Let p be a transition matrix. Prove that all eigenvalues of p are inside
the disk cfw_x + iy : x2 + y 2 1.
Proof. Assume that is an eigenvalue and v is an associated eigenvector. We may assume without loss of generality that |v1 | > |vj | for
j = 1. Fr

Problem 1. Suppose that U is a uniform distribution on (0, 1). Let be the density of
x
standard normal. Let f be the inverse function of F (x) = (y)dy. Show that f (U ) is
a standard normal random variable. In other words, as long as we know how to sample

1. Let p be a transition matrix. Prove that all eigenvalues of p are inside
the disk cfw_x + iy : x2 + y 2 1.
2. For any r [1, 1], nd a 2 2 transition matrix whose eigenvalues
are 1 and r.
3. Consider a nite state Markov chain with state space S. Let A S.

2.27
Proof. Let U Unif(0, 1) be the arrival time of the bus and N (t) be a Poisson process with
rate 6, which indicates how many cars drive by the bus stop up to time t. Then
1
P (N (u) = n none of these n cars can take prof.|U = u)du
P (Prof. rides with

IE 5531 Midterm #1 Solutions
Prof. John Gunnar Carlsson
October 10, 2011
This exam has 8 pages and a total of 5 problems. Make sure that all pages are present. To
obtain credit for a problem, you must show all your work. if you use a formula to answer a p

IE 5531 Midterm - Oct 2013
Page 1 of 9
M IDTERM E XAM
IE 5531
Oct 21st, 2013
INSTRUCTIONS
a) Write ALL your answers in this exam paper.
b) One piece of note is allowed. No computer or cell phone is allowed.
c) The examination time is 10:15am - 12:15pm.
d)

IE 5531 Midterm - Oct 2013
Page 1 of 9
M IDTERM E XAM
IE 5531
Oct 21st, 2013
INSTRUCTIONS
a) Write ALL your answers in this exam paper.
b) One piece of note is allowed. No computer or cell phone is allowed.
c) The examination time is 10:15am - 12:15pm.
d)

5.4
Solution: Let T = infcfw_n 0 : Xn 0.9. Since (Xn ) is a martingale, EXT n = EX0 = 1/2.
Now since XT n 0.9 if T n, we have that
0.9P (T n) 1/2,
which gives P (T n) 5/9. Letting n tend to innity gives
P (Xn 0.9 for some n)P (T < ) 5/9.
5.5
Solution: Let