Question

For $n \geq 2$, let $X_1, X_2, \ldots, X_n$ be a random sample from a distribution with $E(X_1) = 0$, $\text{Var}(X_1) = 1$, and $E(X_1^4) < \infty$. Let $\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$ and $S_n^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X}_n)^2.$
Then which of the following statements is/are always correct?

• A. $E(S_n^2) = 1$ for all $n \geq 2$
• B. $\sqrt{n} \bar{X}_n \xrightarrow{d} Z$ as $n \to \infty$, where $Z$ has the $N(0, 1)$ distribution
• C. $\bar{X}_n$ and $S_n^2$ are independently distributed for all $n \geq 2$
• D. $\frac{1}{n} \sum_{i=1}^n X_i^2 \xrightarrow{P} 2$ as $n \to \infty$

Answer 1

Answer: (A), (B)

$E(S_n^2) = 1$ for all $n \geq 2$

Answer 2

The correct option is (A): $E(S_n^2) = 1$ for all $n \geq 2$,(B): $\sqrt{n} \bar{X}_n \xrightarrow{d} Z$ as $n \to \infty$, where $Z$ has the $N(0, 1)$ distribution