Consider a system with one component that is subject to failure, and suppose tha

Question

Consider a system with one component that is subject to failure, and suppose that we have 115 copies of the component. Suppose further that the lifespan of each copy is an independent exponential random variable with mean 25 days, and that we replace the component with a new copy immediately when it fails.

Approximate the probability that the system is still working after 3500 days.

Probability

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 4125 days.

Probability

Explanation / Answer

a) let lifespan variable is X

for 115 component ; mean =115*25 =2875

as std deviation of exponential dist. =mean

and std deviation =25*(115)1/2=268.0951

hence P(X>3500) =P(Z>(3500-3875)/ 268.0951)

= P(Z>-1.3988)

0.919057

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

and std deviation of replacement time Y= (0.5-0)/(12)1/2 =0.1443

hence total mean time for 115 components =115*25+114*0.25 =2903.5

also std deviation =(115*25^2+114*0.1443^2)^(1/2)=268.1

hence P(X>4125) =P(Z>(4125-2903.5)/268.1)

=P(Z>4.5561) =1-.99999 = 0.00000

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