2. Let X1, X2, ..., X, be a random sample from an exponential distribution Exp(A). (a) Show that the mgf of X Exp(A) is Mx(t) = (1 – At)-1, (b) Let the r.v. Y = ?E, X;. Find the mgf of Y and deduce that Y ~ x(2n). 2 n (c) Derive a 100(1 – a)% CI for A. (d) A random sample of 15 heat pumps of a certain type yielded the following obser- vations on lifetime (in years): 2.0 1.3 6.0 1.9 5.1 0.4 1.0 5.3 15.7 0.7 4.8 0.9 12.2 5.3 0.6 Assume that the lifetime distribution is exponential. What is a 95% CI for the true average and the standard deviation of the lifetime distribution?

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A: Inverse property

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