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CEE 304 – Section 3 Problems
1. Suppose the diameters of ball bearings are normally distributed. You are told that the
diameter of a ball bearing must be within 3.0
±
0.01 cm. No bearing outside this range is
acceptable. You know from process that diameters have
μ
= 3.0 cm and
σ
= 0.005 cm.
a.) On average, how many will be scrapped?
Answer
:
4.56%
b.) How many bearings fall within
±
1 standard deviation of the mean?
Answer
:
68.3%
c.) What diameters delineate the 95
th
and 5
th
percentile?
95
th
percentile =
Z
0.05
, [
Φ
(Z
0.05
) = 0.95] ==> from table, Z
0.05
= 1.645
=> X
=
μ
+
1.645
σ =
3.0082
By symmetry,
5
th
percentile =  Z
0.05
= 1.645
==> X
=
μ −
1.645
σ =
2.9918
2. Suppose the weekly demand for propane gas (in thousands of gallons) is a random
variable X with pdf:
2
1
() 21
fx
x
⎛
=−
⎜
⎝⎠
⎞
⎟
for
1
≤
x
≤
2,
and 0 otherwise
a.) Show that it is a pdf.
First must check if
f
(
x
) is defined for all
x
, which it is!
Second, check if
f
(
x
)
≥
0 for all
x
, which it is!
Third, check if
2
1
()
21
1
f
xd
x
d
x
x
+∞
+∞
−∞
−∞
⎛⎞
⋅=
−
⎜⎟
∫∫
, which it does!
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 Fall '08
 Stedinger

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