(b) (7%) Find the MLE of β, and show that it is an unbiased estimator of β.

C. (8%) Feller Weak Law of Large Numbers Suppose that $X_1, X_2, \ldots, X_n$ are iid random variables and define $S_n = \sum_{j=1}^n X_j$. If $$\lim_{x \to \infty} x P[|X_1| > x] = 0,$$ then $$\frac{S_n}{n} - E[X_1 1_{[|X_1| \leq n]}] \xrightarrow{p} 0 \text{ as } n \to \infty,$$ where $1_{[\cdot]}$ is an indicator function. 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.

(b) (7%) Find the MLE of \(\beta\), and show that it is an unbiased estimator of \(\beta\).

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