B. Suppose that the random variables Y_1, Y_2, , Y_n satisfy Y_i = β x_i + _i, i = 1, 2, , n, where x_1, x_2, , x_n are fixed constants, and _1, _2, , _n are iid N(0, σ^2) with σ^2 unknown. (a) (7%) Find a two-dimensional sufficient statistic for (β, σ^2).

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.

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