The head of an engine is held in place by 6 bolts. The engine will fail if any o
ID: 3220696 • Letter: T
Question
The head of an engine is held in place by 6 bolts. The engine will fail if any of the bolts fail that is, there is no redundancy in the system. Suppose that each bolt has an exponentially distributed lifetime with mean 5 years and assume that bolts fail independently (even though this is a questionable assumption). Let Ai be the event that the i th bolt lasts at least t years and let X be the random variable representing the time at which the engine fails due to head bolt failure.
(a) What is the event {X > t} equivalent to, in terms of the Ai s?
(b) Using the independence of the Ais, compute P[X > t]. Then obtain the pdf and the cdf for X. What distribution does X have?
Explanation / Answer
a) Ai be the event that the i th bolt lasts at least t years
X be the random variable representing the time at which the engine fails due to head bolt failure
P(X> t) = P(min (A1,A2, ...A6) )
b) P(X >t) = P(min (X1,X2,X3...X6 ) >t)
=P(X1 >t)P(X2 >t) ...P(X6 >t)
= ( e^(-t/5))^6
=e^(-6t/5)
cdf = 1 -P(X>t) = 1- e^(-6t/5)
pdf = d/dt (cdf) = e^(-6t/5)
hence X has exponential distribution with mean 5/6 year
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