# Exam P Practice Problem 2

Problem 2a
The lifetime of an electronic device follows an exponential distribution. The device is sold with a one-year warranty. The manufacturer estimated that at the expiration of the warranty, about 85% of the devices are still working. For such a device that was recently purchased, what is the probability that it will still be working in 5 years?

Problem 2b
The lifetime of an electronic device follows an exponential distribution. The device is sold with a one-year warranty. The manufacturer estimated that at the expiration of the warranty, about 90% of the devices are still working. An extended warranty for an additional 2 years is available, covering any needed repair during this period. What proportion of the devices sold will still be working at the end of the extended warranty period?

Solution is found below.

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Solution to Problem 2a
The survival function for these devices is $S(t)=e^{-\alpha t}$. Since 85% of the devices are still working at the end of the warranty period, we have:

$\displaystyle e^{-\alpha}=0.85 \ \ \ \ \ \Longrightarrow \ \ \ \ \ \alpha=ln(0.85)$.

Thus $S(5)=e^{-5ln(0.85)}=0.85^5=0.4437$