Applied Statistics and Probability for Engineers

Also, the following result can be obtained.

134 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

If Xhas a Weibull distribution with parameters and ,

(4-24)

    E 1 x 2  a 1

1



b and 2 V 1 x 2 ^2  a 1

2



b^2 c^ a 1

1



bd

2

 

EXAMPLE 4-25 The time to failure (in hours) of a bearing in a mechanical shaft is satisfactorily modeled as a

Weibull random variable with Determine the mean time until

failure.

From the expression for the mean,

Determine the probability that a bearing lasts at least 6000 hours. Now

Consequently, only 33.4% of all bearings last at least 6000 hours.

P 1 x 60002  1 F 160002 expca

6000

5000

b

(^1)  2
de1.0950.334
E 1 X 2  5000  (^31
11) 0.5 24  5000  334  5000 2 !10,000 hours
 (^1) 2, and 5000 hours.
Figure 4-27 Weibull probability density functions
for selected values of and . 
0
0.0
0.2
0.4
0.6
0.8
1.0
3691215
x
f (x)
1
3.4
4.5
1
2
6.2
δ β
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