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Assume that the life of a roller bearing follows a Weibull

Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger ISBN: 9781118539712 55

Solution for problem 156E Chapter 4.10

Applied Statistics and Probability for Engineers | 6th Edition

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Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger

Applied Statistics and Probability for Engineers | 6th Edition

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Problem 156E

Problem 156E

Assume that the life of a roller bearing follows a Weibull distribution with parameters β = 2 and δ = 10,000 hours.

(a) Determine the probability that a bearing lasts at least 8000 hours.

(b) Determine the mean time until failure of a bearing.

(c) If 10 bearings are in use and failures occur independently, what is the probability that all 10 bearings last at least 8000 hours?

Step-by-Step Solution:
Step 1 of 3

Solution 156E

Step1 of 4:

Let us consider a random variable X it presents the lifetime of a bearing. And X follows weibull distribution with parameters and

Here our goal is:

a). We need to determine the probability that a bearing lasts at least 8000 hours.

b). We need to determine the mean time until failure of a bearing.

c). We need to find the probability that all 10 bearings last at least 8000 hours.


Step2 of 4:

a).

We know that cumulative distribution function of weibull distribution is:

 

Consider,

         

                   

         

     

     

Therefore, P(X > 8000) = 0.5273.


Step3 of 4:

b).

The mean of the weibull distribution is:

               

         

           

                   

                                      [therefore ]

 

                                                   

Therefore, The mean of the weibull distribution is 8862.3.


Step4 of 4:

c).

Here we have n = 10 and from part (a) we have p = 0.5273.

Now,

 

                   

         

  

Therefore, P(X = 10) = 0.00166.


Step 2 of 3

Chapter 4.10, Problem 156E is Solved
Step 3 of 3

Textbook: Applied Statistics and Probability for Engineers
Edition: 6
Author: Douglas C. Montgomery, George C. Runger
ISBN: 9781118539712

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Assume that the life of a roller bearing follows a Weibull