Exponential distribution was used to model the lengths of CD-ROM drives in a two drive system. The two CD-ROM drives operate independently, and at least one drive must be operating for the sytem to eperate successfully. Both drives have a mean length of life of 25,000 hours.
A. The reliability R(t) of a singel CD-ROM drive is the probability that the life of the drive exceeds t hours. Give a formula for R(t).
B. Use the result from part A to find the probabiity that the life of the single CD-ROM drive exceeds 8,760 hours
A. To find the reliability R(t), which is the probability that the life of a single CD-ROM drive exceeds t hours, we can use the cumulative distribution function (CDF) of the exponential distribution.
The CDF of the exponential distribution is given by:
F(t) = 1 - e^(-λt)
Where λ is the rate parameter for the exponential distribution.
For a single CD-ROM drive, we are given that the mean length of life is 25,000 hours. The rate parameter λ can be calculated as the reciprocal of the mean, i.e., λ = 1/25,000.
Substituting this value into the CDF formula, we get:
R(t) = 1 - e^(-t/25,000)
So the formula for the reliability R(t) is R(t) = 1 - e^(-t/25,000).
B. To find the probability that the life of a single CD-ROM drive exceeds 8,760 hours, we can substitute t = 8,760 into the reliability formula we derived in part A.
R(8,760) = 1 - e^(-8,760/25,000)
Calculating this expression will give us the desired probability.