The fan in a particular laptop used by a small department at Smartee University

Question

The fan in a particular laptop used by a small department at Smartee University has an average lifespan of 12,000 hours. 1. a) b) c) d) Identify the distribution and parameter(s) of a fan's lifespan. What is the support? What is the probability that a fan in a laptop will break before its average lifespan? What is the probability that a fan in a laptop operates over 8,600 hours? Given a fan has already operated for 10,000 hours, what's the probability that it will operate more than 14,500 hours total? Given a fan operates no more than 15,000 hours, what's the probability that it breaks before 9,500 hours? e) f) What is the median of a fan's lifespan?

Explanation / Answer

a) here fan's lifespan follows exponential distribution with parameter

beta =12000 =1/lambda; its support is from 0 to infinity

b) probability that a fan in a laptop will break before its

average lifespan =P(X<12000)=1-e-x/ =1-e-12000/12000

=1-e-1 =0.6321

c) probability that a fan in a laptop operates

over 8,600 hours =P(X>8600)=1-P(X<8600)=1-(1-e-8600/12000)

=e-8600/12000 =0.4884

d) probability that it will operate more than 14,500 hours given already operated for 10,000 hours

=P(X>14500|X>10000)=P(X>14500 and X>10000)/P(X>10000)=P(X>14500)/P(X>10000)

=e-14500/12000/e-10000/12000 =0.6873

e) probability that it breaks before 9,500 hours given operates no more than 15,000 hours

=P(X<9500|X<15000) =P(X<9500 and X<15000)/P(X<15000)=P(X<9500)/P(X<15000)

=(1-e-9500/12000)/(1-e-15000/12000) =0.7665

f) for median x:P(X<x)=0.5

0.5 =1-e-x/12000

e-x/12000 =0.5

taking log and solving

x =-12000*ln(0.5)=8317.77 Hours

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