Censored Data a common problem in industry is life testing of components and systems. In this problem, we will assume that lifetime has an exponential distribution with parameter λ, so is an unbiased estimate of μ. When n components are tested until failure and the data X1, X2, .., Xn represent actual lifetimes, we have a complete sample, and is indeed an unbiased estimator of μ. However, in many situations, the components are only left under test until r = n failures have occurred. Let Y1 be the time of the first failure, Y2 be the time of the second failure, , and Yr be the time of the last failure. This type of test results in censored data. There are n



(a) Show that μ = Tr/R is an unbiased estimator for μ. [Hint: You will need to use the memory less property of the exponential distribution and the results of Exercise 7-68 for the distribution of the minimum of a sample from an exponential distribution with parameter.]
(b) It can be shown that V(Tr/r) = 1/λ2r). How does this compare to V1X in the uncensored experiment?

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T, = E Y, + (n - r)Y,