7. (MoM, MLE, FI, DM) Let $X_1, X_2, \dots, X_n$ be a random sample of $Beta(1, \theta)$ random variables. The density
function
$f_X(x|\theta) = c(\theta)(1 - x)^{\theta - 1}$, $0 \le x \le 1$, $\theta > 0$.
(a) Find the value of $c(\theta)$ so that $f_X(x|\theta)$ is a density function.
(b) Find the maximum likelihood estimator for $\theta$.
(c) Find the method of moments estimator of $\theta$.
(d) Use the delta method to estimate the standard deviations of the two estimators.
(e) Find the Fisher information. How do the standard deviation estimates in the previous part relate to the
Fisher information?