The lifetime in hours of a certain component of a machine has the continuous probability density function f (x) = 1 e-/1000 1000 x 2 0. The machine contains five similar components, the lifetime of each having the above distribution. The makers are considering offering a guarantee that not more than two of the original components will have to be replaced during the first 1000 hours of use. Find the probability that such a guarantee would be violated, assuming that the components wear out independently, and that if a component does fail then the replacement used is of particularly high quality and will certainly last for the 1000 hours.

A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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The lifetime in hours of a certain component of a machine has the continuous probability
density function
f (x) =
1
-x/1000
x 2 0.
1000
The machine contains five similar components, the lifetime of each having the above
distribution. The makers are considering offering a guarantee that not more than two of the
original components will have to be replaced during the first 1000 hours of use. Find the
probability that such a guarantee would be violated, assuming that the components wear out
independently, and that if a component does fail then the replacement used is of particularly
high quality and will certainly last for the 1000 hours.
Transcribed Image Text:The lifetime in hours of a certain component of a machine has the continuous probability density function f (x) = 1 -x/1000 x 2 0. 1000 The machine contains five similar components, the lifetime of each having the above distribution. The makers are considering offering a guarantee that not more than two of the original components will have to be replaced during the first 1000 hours of use. Find the probability that such a guarantee would be violated, assuming that the components wear out independently, and that if a component does fail then the replacement used is of particularly high quality and will certainly last for the 1000 hours.
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