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?

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.