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
2x
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
2x
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
80 2. FUNDAMENTAL RELIABILITY MATHEMATICS
0 0.5 1 1.5 2 32.5
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
x -A Lognormal Distributed Variable
f (z)-Probability Density
Function
µ
x
= 0, σ
x
= 0.3
µ
x
= 0, σ
x
= 0.6
µ
x
= 0, σ
x
= 1
Figure 2.23: Plots of PDFs of lognormal distributions.
be directly calculated by the sample data of X.
ln x
and
ln x
, which are the mean and standard
deviation of normal distributed variable ln X, can be determined by using the sample data of
ln X or calculated by using
x
and
x
.
If
x
and
x
are known, we can calculate the mean and standard deviation of the normally
distributed variable ln X :
ln x
D ln
2
x
p
2
x
C
2
x
!
;
ln x
D
s
ln
1 C
2
x
2
x
: (2.102)
If
ln x
and
ln x
are known, we can calculate the mean and standard deviation of the lognormally
distributed variable X :
x
D exp
ln x
C
2
ln x
2
;
x
D
q
exp
2
ln x
1
exp.2
ln x
C
2
ln x
/: (2.103)
In Microsoft Excel, the functions for calculating the PDF and CDF of a lognormal distribution
are:
f
.
x
/
D LOGNORM:DIST
.
x;
ln x
;
ln x
; false
/
(2.104)
F
.
x
/
D P
.
X x
/
D LOGNORM:DIST
.
x;
ln x
;
ln x
; true
/
: (2.105)
In MATLAB, the commands for calculating the PDF and CDF of a lognormal distribution
are:
f
.
x
/
D lognpdf
.
x;
ln x
;
ln x
/
(2.106)
F
.
x
/
D P
.
X x
/
D logncdf
.
x;
ln x
;
ln x
/
: (2.107)
2.12. SOME TYPICAL PROBABILITY DISTRIBUTIONS 81
We can also use the standard normal distribution table to calculate the CDF of a lognormal
distributed variable:
F
.
x
/
D P
.
X x
/
D P
.
ln X ln x
/
D ˆ
ln x
ln x
ln x
: (2.108)
Example 2.43
e life of brakes follow a lognormal distribution with a mean
x
D 60001:4 (miles) and a
standard deviation
x
D 9100:6 (miles), respectively. Use Excel, MATLAB, and the standard
normal distribution table to calculate the probability P .X > 40;000 miles/.
Solution:
Per Equation (2.102), we have:
ln x
D ln
2
x
p
2
x
C
2
x
!
D ln
60001:4
2
p
9100:6
2
C 60001:4
2
!
D 10:9908
ln x
D
s
ln
1 C
2
x
2
x
D
v
u
u
t
ln
1 C
9100:6
2
60001:4
2
!
D 0:1508:
By using the Excel function per Equation (2.105), we have:
P
.
X > 40000
/
D 1 P
.
X 40000
/
D 1 LOGNORM:DIST
.
40000; 10:9908; 0:1508; t rue
/
D 0:9955:
By using the MATLAB command per Equation (2.107), we have:
P
.
X > 40000
/
D 1 P
.
X 40000
/
D 1 logncdf
.
40000; 10:9908; 0:1508
/
D 0:9955:
Per Equation (2.108), we have:
P
.
X > 40000
/
D 1 P
.
X 40000
/
D 1 ˆ
ln x
ln x
ln x
D 1 ˆ
ln 40000 10:9908
0:1508
D 1 ˆ
.
2:6138
/
:
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset