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)