1. Suppose the random sample $X_1, X_2, \dots, X_n$ has the following pdf:
$f(x) = \sqrt{\theta}x^{\sqrt{\theta}-1}$ for $0 < x < 1$, and 0 otherwise.
What is the maximum likelihood estimator (MLE) of $\theta$?
A: $\frac{(\sum_{i=1}^{n} X_i)^2}{n^2}$
B: $\frac{(\sum_{i=1}^{n} \log X_i)^2}{n^2}$
C: $\frac{n^2}{(\sum_{i=1}^{n} \log X_i)^2}$
D: $\frac{\sum_{i=1}^{n} X_i}{n}$
E: $\frac{n^2}{(\sum_{i=1}^{n} X_i)^2}$
