SOLUTION: The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours. (a) What is the probability that an assembly on test fails in les

Algebra ->  Finance -> SOLUTION: The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours. (a) What is the probability that an assembly on test fails in les      Log On


   

Question 1183442: The lifetime of a mechanical assembly in a vibration test is exponentially
distributed with a mean of 400 hours.
(a) What is the probability that an assembly on test fails in less than 100 hours?
(b) What is the probability that an assembly operates for more than 500 hours
before failure?
(c) If an assembly has been on test for 400 hours without a failure, what is the
probability of a failure in the next 100 hours?
(d) If 10 assemblies are tested, what is the probability that at least one fails in less
than 100 hours? Assume that the assemblies fail independently.
(e) If 10 assemblies are tested, what is the probability that all have failed by 800
hours? Assume that the assemblies fail independently.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The pdf of the exponential distribution is f%28x%29+=+lambda%2Ae%5E%28-lambda%2Ax%29 for x+%3E=0 and f(x) = 0 whenever x < 0.
The mean of the distribution is 1%2Flambda.
The cdf is equal to %281%2F400%29%2Aint%28+e%5E%28-t%2F400%29%2C+dt%2C+0%2C+x%29+=+1-e%5E%28-x%2F400%29.
(a) What is the probability that an assembly on test fails in less than 100 hours?
Ans. %281%2F400%29%2Aint%28e%5E%28-x%2F400%29%2C+dx%2C0%2C+100%29+=+1+-+e%5E%28-1%2F4%29+=+0.2212
(b) What is the probability that an assembly operates for more than 500 hours
before failure?
Ans.
(c) If an assembly has been on test for 400 hours without a failure, what is the probability of a failure in the next 100 hours?
Ans.

(d) If 10 assemblies are tested, what is the probability that at least one fails in less than 100 hours? Assume that the assemblies fail independently.

(e) If 10 assemblies are tested, what is the probability that all have failed by 800
hours? Assume that the assemblies fail independently.
Ans.