1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.262 – Discrete Stochastic Processes
Problem Set #8
Issued: April 10, 2010
Due:
April 16, 2010
Reading :
For this problem set, please study all of Sections 5.1 – 5.5.
For the week of April 12  16, begin reading Chapter 6.
1)
Exercise 5.1. (Assume the Markov chain is an arbitrary finite or countably infinite chain.
Do the proof first for n = 1, then for n = 2, and then proceed by induction on n. Also does the
limit,
lim(
( ))
ij
n
F
n
→∞
, necessarily always exist? Either show that it does, or else give a counterexample.)
2) a)
Consider the Markov chain in Figure 5.1. Show that the solutions F
ij
(
∞
) given in Exercise 5.2
satisfy eq. (5.9) for this chain as well. For the special case p = q = 1/2, show that the chain is
recurrent, (i.e., show that the solutions given in Exercise 5.2, F
ij
(
∞
) = 1,
i,j
∀
, are correct) by showing
that there is no smaller solution to eq. (5.9). (You may use, without proof, the symmetric behavior of
the chain in this case, e.g., F
01
(
∞
) = F
10
(
∞
), F
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Moon,J
 Computer Science, Electrical Engineering, Probability theory, Stochastic process, Markov chain, Andrey Markov, Xn, fij

Click to edit the document details