2.15. EXERCISES 103
Table 2.20: Emails per minute
X 0 1 2 3 4 5 6 7 8
p(x) 0.005 0.06 0.22 0.31 0.32 0.06 0.015 0.005 0.005
2.26. e number of emails X that is received by an admission per minute has the following
PMF p.x/ (Table 2.20).
(a) Calculate the mean, standard deviation, and coefficient of the variance.
(b) Determine its CDF.
2.27. A town has three sets of water pump systems. It is assumed that the probability of a
water-pump system functioning without failure for one year is 0.95. It is also assumed
that the performance of these three water pump systems can be described by a binomial
distribution.
(a) Determine its PMF.
(b) Determine its CDF.
(c) Calculate its mean, standard deviation.
(d) Calculate the probability of the event, which is at least one water-pump system
functioning without a failure.
2.28. It is assumed that 65% of a household with an installation of a solar heat system will
have a utility bill in reduction by at least one-third.
(a) What is the probability that a utility bill will be reduced by at least one third in
four out of five households with the installation of solar heat systems?
(b) What is the probability that a utility bill will be reduced by at least one third in at
least three out of five households with the installation of solar heat systems?
2.29. If the probability is 0.25 that any person will dislike the taste of a new ice cream, what
is the probability that 6 out of 18 randomly selected persons will dislike it?
2.30. e arrival of cars at a UPS facility is a Poisson process with a mean arrival rate of three
cars per hour.
(a) Determine the probability of the event that there are five arrival cars in two hours.
(b) Determine the probability of the event that there are at least two arrival cars in two
hours.
104 2. FUNDAMENTAL RELIABILITY MATHEMATICS
2.31. e number of defects in a continuous welded joint in an assembly is 0.05 per meter
length. If the number of defects in the weld follows a Poisson distribution. Find the
following.
(a) e probability of finding exactly one defect in a 10-m length of the weld.
(b) e probability of finding at most three defects in a 10-m length of the weld.
(c) e probability of finding at least one defect in a 2-m length of the weld.
2.32. e number of emails received of an admission account can be described as a Poisson
process with a mean value of 6 per hour. Determine the following.
(a) e probability that no email is received in 30 min.
(b) e probability that at least 2 emails is received in 30 min.
2.33. e hardness of a component follows the uniform distribution with the following PDF:
f
H
.h/ D
8
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
:
0 h < 245 .HB/
1
50
245 h 295 .HB/
0 h > 295 .HB/:
(a) Calculate the mean and standard deviation.
(b) Calculate the probabilities: P .100 < H < 260/, P .H < 300/, P .250 < H <
280/.
2.34. e pressure X (psi) on a device is treated as a random variable and can be described by
a uniform distribution. e PDF is:
f
.
x
/
D
8
<
:
1
4000
4500 x 8500
0 otherwise:
(a) Determine the mean and standard deviation.
(b) Determine the CDF.
(c) Calculate the following probability: P
.
X 3000
/
, P .2000 X 7500/, and
P .6000 X/.
2.35. Loading X in lb on a component is a random variable and can be described by a uniform
distribution:
f
.
x
/
D
8
<
:
1
200
50 x 250
0 otherwise:
2.15. EXERCISES 105
(a) Determine the mean and standard deviation.
(b) Determine the CDF.
(c) Calculate the following probability: P
.
X 220
/
, P .0 X 250/, and P .200
X/.
2.36. e normal distribution is widely used in reliability engineering. Use MATLAB to plot
PDFs of three normal distributions: .
1
D 100;
1
D 50/; .
2
D 100;
2
D 5/; .
3
D
50;
3
D 5/.
2.37. For a normal distribution random variable X with
x
D 100 and
x
D 20, calculate
P .0 < X < 150/; P .50 < X < 200/ and P .X < 120/.
2.38. It is assumed that the fatigue life T (cycles) of a component follows a log-normal dis-
tribution. e mean
T
and the standard deviation
T
based on a set of test data are:
T
D 45000 (cycles) and
T
D 3500 (cycles).
(a) Determine parameters
ln T
and
ln T
of the lognormal distribution.
(b) Use Excel to plot the PDF of this lognormal distribution.
(c) Calculate following probabilities: P .T > 30;000/, P .20;000 < T < 40;000/,
P .T < 60;000/.
2.39. Show the PDF and CDF of two-parameters of Weibull distribution.
(a) Use MATLAB to plot the PDFs of Weibull PDFs with the parameters (a) D
100; ˇ D 1:25; (b) D 100; ˇ D 4; and (c) D 10; ˇ D 1:25.
(b) For the Weibull distribution with the distribution parameter D 20; ˇ D 1:5, cal-
culate its mean and standard deviation.
(c) For the Weibull distribution with the distribution parameter D 20; ˇ D 1:5, cal-
culate the probabilities:
P .X < 80/
,
P .X < 40/
,
P .10 < X < 90/
.
2.40. A normally distributed random variable X has a mean 100 and a standard deviation 20.
(a) Calculate the probability P .50 < X < 150/.
(b) Recalculate the probability P .50 < X < 150/ if the mean is kept the same, but the
standard deviation is reduced to 10.
2.41. e fatigue life X (cycles) of the component at a given stress level is a random variable
and can be described by a log-normal distribution. e mean and standard devotion of
the test data are
X
D 55;000 (cycles) and
X
D 6000 (cycles).
(a) Determine the distribution parameters:
ln X
and
ln X
.
(b) Use the Excel function to calculate the probability: P
.
X 40;000
/
.
106 2. FUNDAMENTAL RELIABILITY MATHEMATICS
(c) Use the MATLAB command to calculate the probability: P
.
X 20;000
/
.
2.42. e measured resistance R of a type of resistor manufactured by a company can be
described by a normal distribution with a mean
R
D 120:0./ and a standard deviation
R
D 0:6./.
(a) Use the Excel function to calculate the probability P .118:5 R 121:5/. Explain
the physical meaning of the calculated value.
(b) If there are 200 pieces of this resistor, estimate the number of these resistors which
has the resistance in the range .118:5 R 121:5/.
2.43. Use MATLAB to plot the PDFs and the CDFs of a normal distribution with three
different sets of distribution parameters: (1) D 100 and D 5; (2) D 100 and D
10; and (3) D 100 and D 30. Discuss the effect of standard deviation on a normal
PDF.
2.44. Use Excel to plot the PDFs and CDFs of a normal distribution with three different
sets of distribution parameters: (1) D 10 and D 10; (2) D 50 and D 10; and
(3) D 100 and D 10. Discuss the effect of the mean on a normal PDF.
2.45. e diameter X of a forged crankshaft follows a normal distribution. e mean and
standard deviation of diameters based on the collected data are
x
D 1:200
00
and
x
D
0:005
00
.
(a) If the diameter of the crankshaft is specified as 1:200 ˙ 0:020
00
, calculate the per-
cent of discarded crankshafts by using Microsoft Excel function.
(b) Use the standard normal distribution table to calculate the probability
P
.
1:180 X < 1:205
/
,
(c) Use the MATLAB command to calculate the probability:
P
.
1:200 3
x
X 1:200 C 3
x
/
P
.
1:200 2
x
X 1:200 C 2
x
/
P
.
1:200
x
X 1:200 C
x
/
:
2.46. Use the MATLAB to plot the PDFs and CDFs of a lognormal distribution with three
different sets of distribution parameters: (1)
ln x
D 3:25 and
ln x
D 0:129; (2)
ln x
D
3:25 and
ln x
D 0:516; and (3)
ln x
D 3:25 and
ln x
D 1:125. Discuss the effect of the
standard deviation
ln x
on the lognormal PDF.
2.47. Use Excel to plot the PDFs and CDFs of a lognormal distribution with three different
sets of distribution parameters: (1)
ln x
D 1:25 and
ln x
D 0:129; (2)
ln x
D 3:25 and
ln x
D 0:129; and (3)
ln x
D 5:25 and
ln x
D 0:129. Discuss the effect of the mean
ln x
on the lognormal PDF.
2.15. EXERCISES 107
2.48. e fatigue strength X (ksi) of a material is a random variable and can be described by a
lognormal distribution with the mean
ln x
D 3:25 and the standard deviation
ln x
D
0:189.
(a) Use MATLAB or Excel to plot its PDF and CDF.
(b) Calculate the mean
x
and the standard deviation
x
of the fatigue strength.
(c) Use MATLAB command to calculate probability P
.
4 X 210
/
and
P
.
X 10
/
.
(d) Use Excel to calculate probability P
.
88 X
/
and P
.
X 15
/
.
2.49. Use MATLAB to plot the PDFs and the CDFs of a Weibull distribution with three
different sets of distribution parameters: (1) D 1 and ˇ D 0:5; (2) D 1 and ˇ D 1:0;
and (3) D 1 and ˇ D 5. Discuss the effect of shape parameter ˇ on a Weibull PDF.
2.50. Use Excel to plot the PDFs and the CDFs of a Weibull distribution with three different
sets of distribution parameters: (1) D 0:5 and ˇ D 5; (2) D 5 and ˇ D 5; and (3) D
10 and ˇ D 5. Discuss the effect of scale parameter on the Weibull PDF.
2.51. e ultimate tensile strength X (ksi) of a type of steel is a random variable and follows
a Weibull distribution with parameters ˇ D 20, and D 100.
(a) Calculate the mean and standard deviation of the ultimate strength.
(b) Use the Excel function to calculate the probability: P
.
X 105
/
;
P
.
98 X 102
/
.
(c) Use the MATLAB command to calculate the probability: P
.
20 X 80
/
;
P
.
50 X
/
.
2.52. A two-speed synchronized transfer case used in a large industrial dump truck experi-
ences failure that seems to be well approximated by a two-parameter Weibull distribu-
tion with D 18; 000 km and ˇ D 2:7.
(a) What is the probability P .x > 10;000/?
(b) What is the probability P .x > 24;000/?
2.53. It is suggested that the stress x of a bridge connection was an exponential distribution
with mean value 6 MPa.
(a) e probability P .x < 10/.
(b) e probability P .5 < x < 10/.
2.54. e time to failure T of a system is an exponential distribution with the parameter
D 0:001. Calculate the probability of the system that can work more than 200 (h).
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