Introduction to Stochastic Processes, Spring 2011
Homework #3
Due: Thursday, February 3rd, 2011 in class
Simple Random Walk Problems
1.
Problem of the Points.
A coin is tossed repeatedly, heads turning up with proba
bility
p
on each toss. Player A wins the game if
m
heads appear before
n
tails have
appeared, and player B wins if
n
tails appear before
m
heads. Let
p
mn
be the proba
bility that A wins the game. Set up a difference equation for the
p
mn
. What are the
boundary conditions?
2. Let
S
n
be a simple random walk with
S
0
= 0, i.e.
S
n
=
n
X
i
=1
X
i
where
X
1
, X
2
, X
3
, . . .
are independent random variables with
P
(
X
i
= 1) =
p
and
P
(
X
i
=

1) = 1

p
=
q
.
(a) Note that at odd times
S
n
is necessarily odd, and at even times it is even. Hence
it can only be back at zero at even times. For
n
≥
0, compute
P
(
S
2
n
= 0).
Hint:
In order to be back at zero at time 2
n
, you need to have made as many
“up” steps as “down” steps. What is the probability of that event?
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 Spring '11
 nomane
 Probability, Probability theory, Stochastic process, Markov chain, Random walk, simple random walk

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