Table 1: Area under the standard normal curve II. A. Let X_1, X_2, , X_n be a random sample from a U(0, θ) population. (a) (7%) Consider H_0: θ = θ_0 versus H_1: θ ≠ θ_0 and define X_(n) = _1 ≤ i ≤ n X_i. Show that T^*(X): Reject H_0 if X_(n) > θ_0 or X_(n) ≤ θ_0 α^1/n is a uniformly most powerful (UMP) test of size α.

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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