Let X1, X2, . . . , Xn be a random sample from an exponential distribution Exp(λ). (a) Show that the mgf of X ∼ Exp(λ) is MX(t) = (1 − λt) −1 , t < 1 λ . (b) Let the r. v. Y = 2 λ Pn i=1 Xi . Find the mgf of Y and deduce that Y ∼ χ 2 (2n). (c) Derive a 100(1 − α)% CI for λ. (d) A random sample of 15 heat pumps of a certain type yielded the following observations 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?