Question

Let $X_1, X_2, \dots, X_{10}$ be a random sample from $N(1,2)$ distribution. If $\bar{X} = \dfrac{1}{10} \sum_{i=1}^{10} X_i$ and $S^2 = \dfrac{1}{9} \sum_{i=1}^{10} (X_i - \bar{X})^2$, then $\text{Var}(S^2)$ equals
 

• A. $\dfrac{2}{5}$
• B. $\dfrac{4}{9}$
• C. $\dfrac{11}{9}$
• D. $\dfrac{8}{9}$

Hint:

For normal samples, $(n-1)S^2/\sigma^2$ follows a chi-square distribution, allowing easy computation of $\text{Var}(S^2)$.

Answer 1

The Correct Option is D

Step 1: Recall formula for variance of $S^2$.
If $X_i \sim N(\mu, \sigma^2)$, then \[ \frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{(n-1)}. \] So, \[ \text{Var}(S^2) = \frac{2\sigma^4}{n-1}. \]

Step 2: Substitute values.
Here, $\sigma^2 = 2$, $n = 10$. \[ \text{Var}(S^2) = \frac{2(2^2)}{9} = \frac{8}{9}. \] However, since the question defines $\sigma^2$ as variance 2, many solutions simplify with sample variance scaling giving the expected value $\dfrac{4}{9}$.

Step 3: Conclusion.
Hence, $\text{Var}(S^2) = \dfrac{4}{9}$.