Question

A dry etcher used in semiconductor manufacturing requires several different industrial gases, whose flow is regulated by six mass flow controllers (MFCs) of the same type. The time to failure of these MFCs has been found to be exponentially distributed with mean of 1000 hours.

a) Assuming all MFCs were installed new at the same time and their failures are statistically independent, compute the probability that none of the MFCs will fail before 800 hours have elapsed.

b) Compute the probability that all these MFCs will have failed after 3000 hours have elapsed.

c) Compute the probability that exactly one of the MFCs will have failed by the time 800 hours have elapsed.

Answer 1

Let the random variable Xrepresent the time to failure of 6 MFCs Given that the random variable Xfollows exponential distribution with mean 1000 hours Hence the probability density function of the random variable Xis, elooo x20 x<0 The fomula for the mean of the binomial distribution is, E(X) - 1000 The cumulative distribution function of F(x) for the exponential distribution is given by, Fx(x)-P(Xsx) -1-e1000 Probability that MFCs fail before 800 hours is calculated as follows -800 P(X<800)-1- -08 = 1-0.4493 0.5507 Hence, the probability that none of the MFCs will fail before 800 hours have elapsed is, P(X(0.5507) (0.4493) = (0.4493)‘ = 0.0082

Probability that MFCs will have failed after 3000 hours is calculated as follows P(X > 3000-1-P(X 3000) 3000 11-e1000 1-(1-0.0498) = 0.0498 Hence, the probability that all these MFCs will have failed after 3000 hours have elapsed is, :(00498)5 (0.9502)。 P(X-б)- = (0.0498 ) = 1.52537E-08 0.0000 Probability that MFCs fail before 800 hours is calculated as follows -800 P(X<800)-1- -08 = 1-0.4493 0.5507 Hence, the probability that none of the MFCs will fail before 800 hours have elapsed is, P(X=1)-: | (0.5507)' (0.4493) (1)(0.5507) (0.01831) 0.0605