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
/
.