82 2. FUNDAMENTAL RELIABILITY MATHEMATICS
By using the standard normal distribution table, that is, Table 2.10 and per Equation (2.96), we
have:
P
.
X > 40000
/
D 1 P
.
X 40000
/
D 1 ˆ
.
2:6138
/
D 1
Œ
1 ˆ
.
2:6138
/
D ˆ
.
2:6138
/
ˆ
.
2:61
/
D 0:9955:
2.12.6 WEIBULL DISTRIBUTION
e Weibull distribution is one of the frequently used and versatile distributions. For example,
it can be used to describe materials strength and fatigue life.
3-parameter Weibull distribution: e PDF of a three-parameter Weibull distribution is:
f
.
x
/
D
ˇ
x
ˇ 1
e
x
ˇ
x ; (2.109)
where is a scale parameter, ˇ is a shape parameter, and is a location parameter. All of these
distribution parameters must be larger than zero.
e CDF of a three-parameter Weibull distribution is:
F
.
x
/
D 1 e
x
ˇ
x : (2.110)
e graphs of three-parameter Weibull distribution with D 1, ˇ D 1, D 0, and 5 are shown
in Figure 2.24.
From Figure 2.24, the two PDF graphs are identical, but the second one shifts to the right
by the location parameter D 5. e three-parameter Weibull distribution can especially de-
scribe the distribution of material strength, which has a minimum proof strength. is minimum
proof-strength would be equal to the location parameter. e most frequently used Weibull dis-
tribution is a two-parameter Weibull distribution.
2-parameter Weibull distribution: e PDF of two-parameter Weibull distribution, typi-
cally termed as Weibull distribution, is:
f
.
x
/
D
ˇ
x
ˇ 1
e
x
ˇ
x 0: (2.111)
e CDF of a two-parameter Weibull distribution is:
F
.
x
/
D 1 e
x
ˇ
x 0: (2.112)
2.12. SOME TYPICAL PROBABILITY DISTRIBUTIONS 83
0 1 2 3 4 5 6 7 8 9 10
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
x
f (x)-PDF
β = 2, η = 1, γ = 0 β = 2, η = 1, γ = 5
Figure 2.24: Graphs of three-parameter Weibull distribution.
Graphs of a Weibull distribution with different and ˇ are shown in Figure 2.25. With the
same shape parameter ˇ D 2, the PDF graph shifts to the right and become flattered when
the scale parameters change from D 100 to D 300. With the same scale parameter D 300,
the PDF graph shifts to the left and become flattered when the shape parameters change from
ˇ D 3 to ˇ D 2. With the combination of different scale and shape parameters and ˇ , Weibull
distribution can be used as the type of distribution of lots of different random variables.
0.01
0.008
0.006
0.004
0.002
0
f (x)-PDF
β = 2, η = 100
β = 2, η = 300
β = 3, η = 300
0 100 200 300 400 500 600
x
Figure 2.25: Graphs of Weibull distributions with different parameters.
84 2. FUNDAMENTAL RELIABILITY MATHEMATICS
e mean and standard deviation of a Weibull distribution are:
x
D E
.
X
/
D
1
ˇ
C
1
(2.113)
x
D Var
.
X
/
D
s
2
ˇ
C 1
1
ˇ
C 1
2
; (2.114)
where
.
a
/
is the gamma function
.
a
/
D
R
1
0
e
z
z
a1
dz. e value of the gamma function
can be determined by Microsoft Excel function or MATLAB command.
By Microsoft Excel,
.
a
/
D GAMMA.a/: (2.115)
By MATLAB command,
.
a
/
D gamma.a/: (2.116)
In Microsoft Excel, the functions for calculating the PDF and CDF of a Weibull distribution
are:
f
.
x
/
D WEIBULL:DIST.x; ˇ; ; false/ (2.117)
F
.
x
/
D P
.
X x
/
D WEIBULL:DIST.x; ˇ; ; true/: (2.118)
In MATLAB, the commands for calculating the PDF and CDF of a Weibull distribution are:
f
.
x
/
D wblpdf .x; ; ˇ/ (2.119)
F
.
x
/
D P
.
X x
/
D wblcdf .x; ; ˇ/: (2.120)
Example 2.44
A bearing life X of a gearbox can be described by a two-parameter Weibull distribution with the
scale parameter D 6000 (hours) and the shape parameter ˇ D 0:62. (1) Determine the mean
and standard deviation of the bearing life. (2) Determine the probability P
.
X > 3000
/
.
Solution:
1. e mean and standard deviation.
Per Equations (2.113) and (2.115) or (2.116), the mean of the bearing life will be:
x
D
1
ˇ
C 1
D 6000
1
0:62
C 1
D 6000
.
2:1629
/
D 6000 1:0802 D 6481:2 .hours/:
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