2. A Battery Charge (25 points) Mendel works as a consultant, and needs a battery to...

Transcribed Image Text:

2. A Battery Charge (25 points) Mendel works as a consultant, and needs a battery to power his laptop so he can work during airplane trips. The brochure about the battery he uses says that the average lifespan of the battery is 60 hours (= 1/A); what it does not say, however, is that the lifespan if exponentially distributed with that mean. Because it's traumatic for Mendel if the battery dies during a flight (leaving him unable to work) he replaces a battery when it reaches age 60 (hours) even if it is still working. Of course, if the battery dies before 60 hours, he suffers a trauma and gets a new battery as soon as he lands. Answer the following questions: (a) Does Mendel's 60-hour rule make sense to you (with full knowledge)? (5 points) (b) What is the probability that a battery will die before it reaches age 60? (5 points) (c) If Mendel has just installed a new battery, how long will it operate on average before he replaces it? (10 points) Hint: First define the following random variables and corresponding events: X is the battery's time until replacement. A is the event that the battery dies before 60. A2 is the event that the battery is still alive at age 60. Now note that A1 and A2 partition the sample space 2 and use the conditional expecta- tion formula. (You will need to compute E[X | A] and there is a clever way to do this by letting Z be the battery's lifespan and noting that E[Z] = 60. But an alternative expression for E[Z] can be obtained via the conditional expectation formula and then comparing E[Z | A] and E[X | A].) (d) By what percentage would his long-term cost of buying batteries go down if he only replaced batteries when they died? (5 points) 2. A Battery Charge (25 points) Mendel works as a consultant, and needs a battery to power his laptop so he can work during airplane trips. The brochure about the battery he uses says that the average lifespan of the battery is 60 hours (= 1/A); what it does not say, however, is that the lifespan if exponentially distributed with that mean. Because it's traumatic for Mendel if the battery dies during a flight (leaving him unable to work) he replaces a battery when it reaches age 60 (hours) even if it is still working. Of course, if the battery dies before 60 hours, he suffers a trauma and gets a new battery as soon as he lands. Answer the following questions: (a) Does Mendel's 60-hour rule make sense to you (with full knowledge)? (5 points) (b) What is the probability that a battery will die before it reaches age 60? (5 points) (c) If Mendel has just installed a new battery, how long will it operate on average before he replaces it? (10 points) Hint: First define the following random variables and corresponding events: X is the battery's time until replacement. A is the event that the battery dies before 60. A2 is the event that the battery is still alive at age 60. Now note that A1 and A2 partition the sample space 2 and use the conditional expecta- tion formula. (You will need to compute E[X | A] and there is a clever way to do this by letting Z be the battery's lifespan and noting that E[Z] = 60. But an alternative expression for E[Z] can be obtained via the conditional expectation formula and then comparing E[Z | A] and E[X | A].) (d) By what percentage would his long-term cost of buying batteries go down if he only replaced batteries when they died? (5 points)