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September 11, 2020

b) Company A produces devices of somewhat limited reliabilit

b) Company A produces devices of somewhat limited reliability. The devices occasionally crash, requiring a reboot. To make matters worse, each device they produce has a different crash rate. For a given device, the time until the next crash is distributed asY∼Exp(1/X), whereXis the expected crash time for the given device and is in turn i.i.d for each device with a known mean and varianceμ,σ2.Now let’s say we take one device made by this company and let it run until it has crashed 5 times.LetY1denote the time until the first crash andY5the time until the fifth crash. What is cov(Y1,Y5)?You may assume that rebooting the device after a crash takes no time.c) Suppose you repeatedly cast a 6-sided die but stop as soon as you cast an odd number. Let S denote the sum of all the throws, excluding the final odd throw (for instance, if you roll an odd number on the first try, thenS= 0). What isE(S) and var(S)?

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