UCSB
Fall 2009
ECE 235:
Problem Set 5
Assigned:
Friday November 13
Due:
Tuesday November 24 (by noon, in course homework box)
Reading:
Hajek, Chapter 4 and selected parts of Chapter 6; class notes
Topics:
Independent increments processes; Markov processes
Practice problems (not to be turned in):
The even numbered problems in Chapter 4, up to
Problem 4.34. Ignore all problems (or parts of problems) related to martingales, since we have not
covered these. Compare your solutions with the solutions provided to make sure you understand the
concepts.
Problem 1:
Consider the random walk
X
n
=
W
1
+
...
+
W
n
+
X
0
, with
{
W
k
}
i.i.d. with
P
[
W
k
=
+1] =
p
= 1

P
[
W
k
=

1]. Set
X
0
= 0. Suppose that
p <
1
/
2, so that the random walk is drifting
downwards. We want to ±nd the probability that the random walk hits a positive level (which should
become more and more unlikely, the higher the level).
(a) For any positive integer
k
, let
p
k
=
P
[
max
n
≥
1
X
n
≥
k
]. Show that
p
k
=
p
k
1
.
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 Fall '08
 Brewer,F
 Poisson Distribution, Probability theory, Stochastic process, Exponential distribution, Poisson process, Markov chain

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