C. (8%)
Let F be a symmetric distribution and suppose
                                        $1 - F(x) = \frac{e}{2x \ln x}, \quad x \geq e.$
Suppose that $\{X_n, n \geq 1\}$ is an iid sequence of random variables with comment distribution F. Show that the sample mean converges in probability to 0.

(10%) Let \( X \) and \( Y \) denote independent random variables with respective probability density function \( f(x) = 2x, 0 < x < 1 \), zero elsewhere, and \( g(y) = 3y², 0 < y < 1 \), zero elsewhere. Let \( U = \min(X, Y) \) and \( V = \max(X, Y) \). Find the joint probability density function of \( U \) and \( V \).

(a) (10%) Find the maximum likelihood estimator \( \hat{\theta} \) of \( \theta \) and the distribution of \( \hat{\theta} \).

(b) (5%) Find \( \xi{p} \) and the corresponding maximum likelihood estimator \( \hat{\xi}{p} \) of \( \xi_{p} \).

(c) (10%) Find the asymptotic distribution of \( \xi_{p} \).

(d) (10%) Let \( 0 < X{(1)} \leq X{(2)} \leq \dots \leq X{(n)} \) denote the order statistics \( X{1}, X{2}, \ldots, X{n} \). Find \( E[F(X{(k)})] \), where \( k \doteq p(n + 1) \). Based on the result, find another estimator \( \hat{\xi}{p} \) of \( \xi_{p} \)?

(e) (5%) Suppose a random sample of size 49 is collected as follows:0.06 0.06 0.09 0.15 0.17 0.17 0.18 0.21 0.23 0.29 0.29 0.31 0.34 0.34 0.370.41 0.41 0.42 0.44 0.45 0.46 0.47 0.47 0.49 0.49 0.49 0.50 0.53 0.56 0.570.57 0.58 0.58 0.59 0.59 0.60 0.63 0.64 0.66 0.68 0.68 0.71 0.74 0.78 0.810.85 0.85 0.91 0.97.We obtain $\sum{i=1}^{49} xi = 23.84$ and $\sum{i=1}^{49} \log(xi) = -43.052$. Find $\hat{\xi}{0.6}$ and $\hat{\xi}{0.6}$.

1. (10%) Derive the Maximum Likelihood Estimator (MLE) of \( \theta \).

2. (10%) Is your MLE unbiased? Prove or disprove the unbiasedness.

3. (10%) Calculate the variance of your MLE. Compare it to the Cramér-Rao Lower Bound and comment on your observation.

(10%) Use your MLE to conduct a hypothesis testing of \[ H_0: \theta = 2 \text{ versus } H_1: \theta > 2 \text{ .} \]Given that the size of the test is 0.05, find the rejection region.

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