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ORIE 3500/5500 Fall Term 2008
Assignment 8Solution
1. Suppose that the printer goes down after
T
hours from 9 a.m. Then
f
T
(
t
) =
e

t/
4
/
4 for
t
≥
0.
(a) Let
X
be the indicator random variable
X
=
±
1
if the printer is working at 1 p.m.
0
otherwise
.
Now
E
(
X

T
) = 1 if
T >
4 and for
T
≤
4,
E
(
X

T
) =
Z
4

T
0
e

x
dx
= 1

e

(4

T
)
.
Hence the required probability is
E
(
X
) =
E
[
E
(
X

T
)] =
Z
4
0
[1

e

(4

t
)
]
e

t/
4
4
dt
+
Z
∞
4
1
·
e

t/
4
4
dt.
=
Z
∞
0
e

t/
4
4
dt

(
e

4
/
4)
Z
4
0
e
3
t/
4
dt
= 1

(
e

4
/
4)(4
/
3)(
e
3

1) = 1

(
e

1

e

4
)
/
3 = 0
.
8940
.
(b) Let
X
be the indicator random variable
X
=
±
1
if the printer is working between 1 and 2 p.m.
0
otherwise
.
Now
E
(
X

T
) = 1 if
T >
5,
E
(
X

T
) = 0 if 4
< T
≤
5 and for
T
≤
4,
E
(
X

T
) =
Z
4

T
0
e

x
dx
= 1

e

(4

T
)
.
Hence the required probability is
E
(
X
) =
E
[
E
(
X

T
)] =
Z
4
0
[1

e

(4

t
)
]
e

t/
4
4
dt
+
Z
5
4
0
·
e

t/
4
4
dt
+
Z
∞
5
1
·
e

t/
4
4
dt.
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 Fall '08
 TURNBULL

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