(1 point) Consider a system with one component that is subject to failure, and s

Question

(1 point) Consider a system with one component that is subject to failure, and suppose that we have 120 copies of the component. Suppose further that the lifespan of each copy is an independent exponential random variable with mean 20 days, and that we replace the component with a new copy immediately when it fails. (a) Approximate the probability that the system is still working after 2800 days. Probability (b) Now, suppose that the time to replace the component is a random variable that is uniformly distributed over (0, 0.5). Approximate the probability that the system is still working after 3200 days. Probability

Explanation / Answer

a) let lifespan variable is X

for 120 component ;

mean (mu) =120*20 =2400

as standard deviation of exponential distribution (sigma) = mean (mu)

=> standard deviation = 20*(120)1/2 = 219.09

hence P(X>2800) =P(Z>(2800-2400) / 219.09) = P(Z>1.8257) = 1-P(Z<1.8257) = 1 - 0.9661 = 0.0339

b) here mean of replacement time Y = (0+0.5)/2 = 0.25

and std deviation of replacement time Y= (0.5-0)/(25)1/2 = 0.1

hence total mean time for 90 components = 120*20 + 119*0.25 = 2429.75

also std deviation =(120*202 + 119 * 0.12)1/2 = 219.09

hence P(X>3200) =P(Z>(3200 - 2429.75) / 219.09) = P(Z>3.5157) = 0.0002

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