Life-testing results on a first-generation microprocessor-based(computer-control

Question

Life-testing results on a first-generation microprocessor-based(computer-controlled)
toaster indicate that X, the life-span (in years) of the centralcontrol chip, is a
random variable that is reasonably well-modeled by the pdf:
f(x)=1/(e(-x/), x>0

with = 6.25 . A malfunctioning chip will have to be replacedto restore proper
toaster operation.
a. The warranty for the chip is set at w x years (in wholeintegers) such that no
more than 15% of the chips would have to be replaced before thewarranty
period expires. Find xw.
2
b. In planning for the second generation toaster, design engineerswish to set a
target value to aim for *
= 2 such that 85% of the second generationchips
survive beyond 3 years. Determine *2
and interpret your results in terms of
the implied increase in mean life-span from the first to the secondgeneration of chips.

Explanation / Answer

I'm having trouble with the computer not letting me do thecustom equation, but I'll answer here. Pardon thenotations. We need the lifetime of the chip to exceed some valuexw , with a probability of .85.
This is like finding a median, actually it's the 85thquantile. (Median is the 50th quantile). Intgrate from 0 to xw the pdf(1/6.25)e^-(x/6.25).   After you integrate, and evaluateat the limits, you get: 1 - e^-(xw/6.25), which must evaluate = .85. solve for xw. a couple steps of algebra later... -xw/6.25 =ln .15. then xw = 11.857. The warranty has to cover whole years, so 12 years should doit. Part 2. Now integrate from 0 to 3, the answer still hasto = .85, but we are solving for . I think you should be able to do it, but if not, repost anddrop me a message and I'll come back.

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