# Question: Find the Rao Cram r lower bound and thus the asymptotic variance

Find the Rao–Cramér lower bound, and thus the asymptotic variance of the maximum likelihood estimator θ, if the random sample X1, X2, . . . , Xn is taken from each of the distributions having the following pdfs:

(a) f(x; θ) = (1/θ2) x e−x/θ, 0 < x < ∞, 0 < θ < ∞.

(b) f(x; θ) = (1/2θ3) x2 e−x/θ, 0 < x < ∞, 0 < θ < ∞.

(c) f(x; θ) = (1/θ) x(1−θ)/θ, 0 < x < 1, 0 < θ < ∞.

(a) f(x; θ) = (1/θ2) x e−x/θ, 0 < x < ∞, 0 < θ < ∞.

(b) f(x; θ) = (1/2θ3) x2 e−x/θ, 0 < x < ∞, 0 < θ < ∞.

(c) f(x; θ) = (1/θ) x(1−θ)/θ, 0 < x < 1, 0 < θ < ∞.

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