2.12. SOME TYPICAL PROBABILITY DISTRIBUTIONS 79
Solution:
Per Equations (2.99) and (2.96), we have:
P
.
50 X 70
/
D ˆ
70 61:2
4:25
ˆ
50 61:2
4:25
D ˆ
.
2:07
/
ˆ
.
2:64
/
D ˆ
.
2:07
/
Œ
1 ˆ
.
2:64
/
D ˆ
.
2:07
/
C ˆ
.
2:64
/
1:
From Table 2.10, we have
ˆ
.
2:07
/
D 0:9808; ˆ
.
2:64
/
D 0:9959:
erefore, we have:
P
.
50 X 70
/
D ˆ
.
2:07
/
C ˆ
.
2:64
/
D 0:9808 C 0:9959 1 D 0:9767:
2.12.5 LOG-NORMAL DISTRIBUTION
e lognormal distribution is a widely used type of distributions. Material ultimate strength,
fatigue life, and fatigue strength might follow a lognormal distribution.
Lognormal distribution: A random variable X is a lognormal distribution if its logarithm
ln X follows a normal distribution. e PDF of a lognormal distributed random variable X is:
f
.
x
/
D
1
p
2x
x
exp
"
1
2
x
x
x
2
#
x 0; (2.100)
where
x
and
x
are the mean and standard deviation of the lognormal distributed random
variable X.
e CDF of a lognormal distributed random variable X is:
F
.
x
/
D P
.
X x
/
D
Z
x
0
1
p
2x
x
exp
"
1
2
x
x
x
2
#
dx x 0: (2.101)
e plots of lognormal distributed variables with the same mean
x
D 0 but different standard
deviations
x
D 0:3; 0:6, and 1 are shown in Figure 2.23. e PDF’s shape of a lognormal dis-
tribution is not symmetrical anymore. With a bigger standard deviation, the shape of the PDF
of a lognormal distribution is skewed toward the left.
For a lognormal distributed variable X, its logarithm ln X is a normal distribution.
x
and
x
, which are the mean and standard deviation of the lognormal distributed variable X, can