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IE 300/GE 331 Homework 4 Sol
ASSIGNMENT:
48, 415, 436, 442, 462, 482, 4102, 4180 (from the textbook, 5
th
edition)
Problem Conversion Guide:
Problem number in 5
th
Edition
Corresponding Problem Number in 4
th
Edition
48
46
415
415
436
430
442
436
462
454
482
470
4102
488
4180
4150
48
The probability density function of the time to failure of an electronic component in a copier (in hours) is
g(G) = ±
²
³
´µµµ
/1000
for
G > 0
. Determine the probability that
(a) A component lasts more than 3000 hours before failure.
(b) A component fails in the interval from 1000 to 2000 hours.
(c) A component fails before 1000 hours.
(d) Determine the number of hours at which 10% of all components have failed.
a)
05
.
0
1000
)
3000
(
3
3000
3000
1000
1000
=
=

=
=

∞
∞


∫
e
e
dx
e
X
P
x
x
b)
233
.
0
1000
)
2000
1000
(
2
1
2000
1000
2000
1000
1000
1000
=

=

=
=
<
<




∫
e
e
e
dx
e
X
P
x
x
c)
6321
.
0
1
1000
)
1000
(
1
1000
0
1000
0
1000
1000
=

=

=
=
<



∫
e
e
dx
e
X
P
x
x
d)
10
.
0
1
1000
)
(
1000
/
0
0
1000
1000
=

=

=
=
<



∫
x
x
x
e
e
dx
e
x
X
P
x
x
.
Then,
e
x

=
/
.
1000
09
, and x =

1000 ln 0.9 = 105.36.
415
Determine the cumulative function for the distribution in Exercise 41.
41 Suppose that
g(G) = ±
²¶
for
0 < G
.
Since we have:
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g(G) = {
0 G ≤ 0
±
−G
G > 0
We have
²(G) = ³(´ ≤ G) = µ
g(¶)·¶ = ¸ − ±
¹º
g»¼ G > 0
º
¹½
,
where C is a constant.
Since when x> +∞, F(x)=1, we have C=1, Hence,
²(G) = {
0 G ≤ 0
1 − ±
−G
G > 0
436
The probability density function of the weight of packages delivered by a post office is
g(G) =
70/(69G
¾
)
for
1 < G < 70
pounds.
(a) Determine the mean and the variance of weight.
(b) If the shipping cost is $2.50 per pound, what is the average shipping cost of a package?
(c) Determine the probability that the weight of a package exceeds 50 pounds.
(a)
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This note was uploaded on 01/26/2012 for the course IE IE 300 taught by Professor Zafarani during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 Zafarani

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