Question

The optimal scheduling of preventative maintenance tests of some (but not all) of n independently operating components was developed. The time (in hours) between failures of a component was approximated by an exponentially distributed random variable with mean 1200 hours.

Find the probability that the time between a component failures ranges is at least 1500 hours.

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Find the probability that the time between a component failures ranges between 1500 and 1700 hours.

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Suppose a component is still working after 1500 hours, find the conditional probability that it will fail before 1700 hours.

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Suppose 7 components are tested. What is the probability that 1 of them failed between 1500 and 1700 hours?

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If 7 components are tested, find the probability that at least 1 of them failed between 1500 and 1700 hours

Answer 1

mean β = 1200

1)

probability =

P(X>1500)=

1-P(X<1500)=

1-(1-exp(-1500/1200))=

0.2865

2)

probability =

P(1500<X<1700)=

(1-exp(-1700/1200)-(1-exp(-1500/1200))=

0.0440

3)

P(X<1700|X>1500) =P(1500 <x<1700)/P(X>1500 ) =P(X<200) =1-exp(-200/1200)=0.1535

4)

from binomial distribution:

probability that 1 of them failed between 1500 and 1700 hours =(7C1)*(0.0440)¹(1-0.0440)⁶ =0.2351

5)

probability that at least 1 of them failed between 1500 and 1700 hours

=1-P(none ) =1-(1-0.0440)^7 =0.2702