IE 336
Oct. 8, 2008
Name:
Test #1
1.
Let
f
(
x, y
) =
(
ye

3
x
0
≤
x
≤ ∞
,
√
3
≤
y
≤
c
0
otherwise
be the joint pdf of two continuous random variables
X
and
Y
.
(a)
Determine the value of the constant
c
.
(b)
Are the random variables
X
and
Y
independent? Explain.
(c)
Compute
E
(
X

y
),
E
(
Y

x
), and
E
(
XY
).
1
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IE 336
Oct. 8, 2008
Name:
2.
Let
X
be an ergodic Markov chain with states
{
1
,
2
,
3
}
. Let
P
=
0
.
1
0
.
5
0
.
4
0
.
3
0
.
1
0
.
6
0
.
1
0
.
1
0
.
8
be the onestep transition matrix of the Markov chain
X
. Clearly
p
13
=
P
(
X
1
= 3

X
0
=
1) = 0
.
4,
p
32
=
P
(
X
1
= 2

X
0
= 3) = 0
.
1, etc.
(a)
Determine
P
(
X
1
= 1

X
0
= 2) and
P
(
X
6
= 2

X
5
= 1
, X
4
= 3).
(b)
Compute
P
(
X
7
= 2

X
5
= 3).
(c)
Find the walk probability
p
1321112
.
(d)
Assume that at some point the process is in state 3. Compute the expected number of
visits to state 1 in the following two steps.
(e)
Assume that at time step
k
the process is two times more likely to be in state 1 than in
state 3. Also, assume that at the same time step
k
the process is two times less likely
to be in state 2 than in state 1. Find the probability that at time step
k
+2 the process
is in state 3.
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 Spring '08
 Bruce,S
 Probability theory, Englishlanguage films, Markov chain

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