108 2. FUNDAMENTAL RELIABILITY MATHEMATICS
2.55. e time to failure X of a system is a random variable and can be described by an
exponential distribution with the parameter D 0:002.
(a) Use MATLAB or Excel to plot its PDF and CDF.
(b) Calculate the probability of the system that can work more than 100 (h).
2.56. A set of 200 test data X of ultimate material strength is collected. e mean
x
and
the standard deviation
x
calculated from the collected data are:
x
D 101:53 (ksi) and
x
D 55:37 (ksi). e ten non-overlapping intervals and the observed number of data
in the corresponding intervals are shown in Table 2.21.
Table 2.21: Intervals and observed number of data
Non-Overlapping Interval
o
j
e
j
(o
j
– e
j
)
2
e
j
Lower range x
j-L
Upper range x
j-U
17.10 48.88 22
48.88 80.66 64
80.66 112.44 49
112.44 144.22 30
144.22 176.00 16
176.00 207.78 8
207.78 239.56 3
239.56 271.34 5
271.34 303.12 2
303.12 334.91 1
W =
(o
j
– e
j
)
2
e
j
10
j = 1
Use Table 2.21 and the Excel function to do the following.
(a) Use the
2
test to check the assumption with a significant level a D 0:05 that it
follows a normal distribution.
(b) Use the
2
test to check the assumption with a significant level a D 0:05 that it
follows a lognormal distribution.
(c) Use the
2
test to check the assumption with a significant level a D 0:05 that it
follows a Weibull distribution.
2.15. EXERCISES 109
2.57. A yield strength data of material specimen are listed in Table 2.22. Input the data in the
first column in an Excel file.
Table 2.22: Yield strength
Yield Strength Data
61.1, 68.5, 45.3, 63.0, 59.9, 50.7, 55.7, 60.0, 78.3, 73.7, 50.5, 75.2, 62.2, 57.7, 62.1, 56.9, 57.4, 66.5, 66.1, 66.1,
61.9, 51.3, 62.2, 67.3, 60.9, 63.9, 62.2, 56.4, 59.8, 53.7, 63.1, 51.6, 52.1, 53.5, 41.5, 66.2, 59.9, 53.8, 65.8, 48.4,
57.5, 56.7, 59.9, 59.9, 53.2, 57.9, 57.2, 61.6, 64.3, 64.4, 53.2, 58.5, 51.2, 51.8, 58.1, 66.8, 53.8, 60.2, 56.8, 64.4,
51.9, 58.3, 61.2, 64.3, 66.8, 58.6, 49.7, 53.9, 52.1, 71.4, 54.6, 62.3, 57.0, 63.1, 53.8, 50.2, 50.1, 60.9, 57.1, 57.0,
66.1, 59.7, 59.2, 67.1, 53.6, 62.0, 62.8, 56.7, 59.3, 51.5, 51.6, 58.7, 62.2, 72.7, 54.3, 59.2, 57.6, 47.2, 55.6, 48.0
Create MATLAB programs to do the following.
(a) Build its histogram.
(b) Calculate the mean and standard deviation of the collected data.
(c) Use the
2
test to check the assumption with a significant level a D 0:05 that it
follows a normal distribution.
(d) Use the
2
test to check the assumption with a significant level a D 0:05 that it
follows a lognormal distribution.
(e) Use the
2
test to check the assumption with a significant level a D 0:05 that it
follows a Weibull distribution.
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