Probability for Engineering Applications
SOLUTION Assignment #15
The Central Limit Theorem has infinite applications. Here are four short examples.
1. (2 pts) A fair coin is tossed 1000 times. Estimate the probability that the number of
heads is between 470 and 530.
The number of independent trials is
250
,
500
,
1000
2
=
=
=
=
=
npq
np
m
n
s
.
9428
.
0
0286
.
0
2
1
)
90
.
1
(
2
1
)
90
.
1
(
)
90
.
1
(
250
500
530
250
500
470
530
470
[
=
×

=

=


=

<

<

=
<
<
Q
Q
Q
m
N
P
N
P
s
2. (2 pts) A fair die is tossed 100 times. Estimate the probability that the total number of
spots is between 330 and 380. (First compute the expected number of spots and its
variance for a random toss of the die.) (cf. 4.24)
Let
i
X
be the number of spots on the ith trial
92
.
2
])
[
(
]
[
]
[
167
.
15
6
36
25
16
9
4
1
]
[
5
.
3
6
6
5
4
3
2
1
]
[
2
2
2
=

=
=
+
+
+
+
+
=
=
+
+
+
+
+
=
i
i
i
i
i
X
E
X
E
X
VAR
X
E
X
E
For 100 tosses,
292
]
[
100
]
[
350
]
[
100
]
[
100
100
=
=
=
=
i
i
X
VAR
S
VAR
X
E
S
E
8503
.
0
0357
.
114
.
0
1
)
76
.
1
(
)
17
.
1
(
1
)
76
.
1
(
)
17
.
1
(
292
350
380
292
350
292
350
330
]
380
330
[
100
100
=


=


=


=

<

<

=
<
<
Q
Q
Q
Q
S
P
S
P
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3. (3pts) The cartridge of a heavily used printer has an exponential lifetime with mean
4 weeks. Use the Central Limit Theorem to find out how many cartridges should be
bought at the beginning of a year so that the probability of not running out during the
year is greater than 0.99 (cf. 4.26)
Let X
i
be the lifetime of the
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 Spring '03
 WOODS
 Probability theory, Yi, 2 pts, 4 weeks, 3pts

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