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

The Correct Option is A, B

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