Math 5040-1 Midterm
Fall 2008
Instructions:
- This exam is due Wednesday October 29, in lecture. Late exams are
not accepted.
- You may not discuss this exam with other persons; this includes other
students in this course. Failure to do this will result in a zero in this
exam; further action might also be taken in accord with the university
bylaws. By taking this exam you are agreeing that this represents your
work alone.
Problems:
1. (Theoretical problem) Let X:= {Xn}∞
n=0 denote a Markov chain on a
finite state space Swith transition matrix P, and consider the stochas-
tic process Y:= {Yn}∞
n=0 defined by setting Yk:= X2k. Is Ya Markov
chain? Prove or disprove carefully.
2. (Theoretical problem) Consider a Markov chain X:= {Xn}∞
n=0 on the
state space S:= {1,2,3}with the following transition matrix:
P:=
0 1 0
0 0 1
1 0 0
.
And the initial probability distribution of Xis ¯
φ0:= (1/3,1/3,1/3).
For all integers n≥0 define the random variable Ynto be the indicator
of the event {Xn6= 1}. That is, Yn:= 0 if Xn= 1 and Yn:= 1 other-
wise. Is Y:= {Yn}∞
n=0 a Markov chain? Prove or disprove carefully.
3. (Simulation problem) A 52-card deck of card is thoroughly shuffled;
then all of the cards are layed out in order [from left to right, say].
What is the probability that a King is set immediately next to an Ace?
1