point) Consider a system with one component that is subject to failure, and supp

Question

point) Consider a system with one component that is subject to failure, and suppose that we have 105 coples of the component Suppose further that the ifespan of each copy is an independent exponentiar ndom vaiable 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 athter 3250 days robabiity ) Now, suppose that the time to reptace the component is a random varable that is unitormly distributed over (0,0.5) Approximate the probabity that the system is stl working ater 3875 days te: You can earn partial credt on this orobiem

Explanation / Answer

Solution:

a) let lifespan variable is X

for 105 component ; mean = 105*25 = 2625

as std deviation of exponential dist. = mean

and std deviation = 25*sqrt(105) = 256.1738

hence P(X > 3250) =P(Z > (3250-2625)/256.1738)= P(Z > 2.4397) = 1P (Z < 2.4397 ) = 10.9927=0.0073

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

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

hence total mean time for 105 components = 105*25+104*0.25 = 2651

also std deviation = sqrt(105*25^2+104*0.1443^2) = 256.17

hence P(X > 3675) =P(Z>(3675-2651)/256.17)

=P(Z>3.9973) = 1P(Z<3.9973)=11.0000 = 0.0000

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