Find the Rao–Cramér lower bound, and thus the asymptotic

QUESTION:

Find the Rao–Cramér lower bound, and thus the asymptotic variance of the maximum likelihood estimator \(\hat \theta\), if the random sample \(X_1, X_2, \ldots , X_n\) is taken from each of the distributions having the following pdfs:

(a) \(f(x ; \theta)=\left(1 / \theta^{2}\right) x e^{-x / \theta}, \quad 0<x<\infty, \quad 0<\theta<\infty\).

(b) \(f(x ; \theta)=\left(1 / 2 \theta^{3}\right) x^{2} e^{-x / \theta}, \quad 0<x<\infty, \quad 0<\theta<\infty\).

(c) \(f(x ; \theta)=(1 / \theta) x^{(1-\theta) / \theta}, \quad 0<x<1, \quad 0<\theta<\infty\).