Question Statement
Let $X_1, X_2, \dots, X_n$ be an i.i.d. random sample from $\mathcal{N}(\mu, \sigma^2)$ where $\mu$ and $\sigma^2$ are unknown. Which of the following statements is/are TRUE?
(A)
The sample mean $\bar{X}$ is the Uniformly Minimum Variance Unbiased Estimator (UMVUE) for $\mu$
(B)
The sample variance $S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2$ is an unbiased estimator of $\sigma^2$
(C)
By Basu's theorem, the sample mean $\bar{X}$ and sample variance $S^2$ are statistically independent
(D)
The Maximum Likelihood Estimator (MLE) for $\sigma^2$ is unbiased for all finite sample sizes $n$
Step-by-Step KaTeX Derivation & Detailed Explanation
### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Concept & Domain Context:** * This question evaluates **Fundamentals of Bose-Einstein and Fermi-Dirac statistics** under *Bose-Einstein and Fermi-Dirac statistics* in GATE PH. 2. **Stage 2: Mathematical Derivation:** * Under normality: $\bar{X}$ is complete and sufficient for $\mu$, so by Lehmann-Scheffe it is UMVUE (A is true). Dividing by $n-1$ yields $E[S^2] = \sigma^2$ (B is true). $S^2$ is ancillary for $\mu$, so by Basu's theorem $\bar{X}$ and $S^2$ are independent (C is true). MLE is $\frac{1}{n}\sum (X_i - \bar{X})^2$, which is biased (D is false). 3. **Stage 3: Distractor Elimination Forensics:** * The evaluated correct answer is derived using standard physical equations. * Distractor values correspond to typical unit conversion omissions or inverted formulas. 4. **Stage 4: Common Student Trap:** * *Trap:* Assuming the Maximum Likelihood Estimator for variance is unbiased in finite samples. *Reference: R.K. Pathria, Paul D. Beale, Statistical Mechanics (Academic Press).*