2.12. SOME TYPICAL PROBABILITY DISTRIBUTIONS 85
Per Equations (2.114) and (2.115) or (2.116), the standard deviation of the bearing life
will be:
x
D
s
2
ˇ
C 1
1
ˇ
C 1
2
D 6000
s
2
0:62
C 1
1
0:62
C 1
2
D 6000
q
.
4:2258
/
Œ
.
2:1629
/
2
D 6000
p
8:0243 2:0840
D 14623:64 .hours/:
2. P
.
X > 3000
/
.
Per Equation (2.112), we can directly calculate the probability.
P
.
X > 3000
/
D 1 P
.
X 3000
/
D 1 F
.
x
/
D 1
"
1 e
x
ˇ
#
D e
.
3000
6000
/
0:62
D
0:5217:
By using Microsoft Excel per Equation (2.118), we have:
P
.
X > 3000
/
D 1 P
.
X 3000
/
D 1 WEIBULL:DIST
.
3000;0:62;6000; true
/
D 1 0:4783 D 0:5217:
By using MATLAB per Equation (2.120), we have:
P
.
X > 3000
/
D 1 P
.
X 3000
/
D 1 wbl
.
3000;6000;0:62
/
D 1 0:4783 D 0:5217:
2.12.7 EXPONENTIAL DISTRIBUTION
e exponential distribution is one of the commonly used distributions. It is a simple distribution
and is a special case of a Weibull distribution with the shape parameter ˇ D 1. e exponential
distribution is used to model the behavior of a device with a constant failure rate.
Exponential distribution: It is a one-parameter distribution. e PDF of an exponential dis-
tribution is:
f
.
x
/
D e
x
x 0; > 0; (2.121)
86 2. FUNDAMENTAL RELIABILITY MATHEMATICS
where is the parameter of the distribution, often called the rate parameter.
e CDF of an exponential distribution is:
F
.
x
/
D 1 e
x
x 0; > 0: (2.122)
e graphs of the PDF of an exponential distribution with D 0:5; 1, and 1.5 are shown in
Figure 2.26. e PDF of an exponential distribution has its maximum values at x D 0 and will
be exponentially decreased.
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
f (x)-PDF
λ = 1.5
λ = 1
λ = 0.5
0
1 2 3 4 5 6
x
Figure 2.26: Graphs of the PDF of an exponential distribution with different .
e mean and the standard deviation of an exponential distribution are:
x
D E
.
x
/
D
Z
1
0
xe
x
dx D
1
(2.123)
x
D
p
Var
.
X
/
D
q
E
Œ
X
2
2
x
D
1
: (2.124)
e PDF and CDF of an exponential distribution are easy to be calculated. Microsoft EXCEL
and MATLAB still provide corresponding functions or commands for it.
In Microsoft Excel, the functions for calculating the PDF and CDF of an exponential
distribution are:
f
.
x
/
D EXPON:DIST.x; ; false/ (2.125)
F
.
x
/
D P
.
X x
/
D EXPON:DIST.x; ; true/: (2.126)
In the MATLAB, the commands for calculating the PDF and the CDF of an exponential
distribution are:
f
.
x
/
D exppdf .x; 1=/ (2.127)
F
.
x
/
D P
.
X x
/
D expcdf .x; 1=/: (2.128)
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