Bessel's Correction and Unbiased Sample Variance

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Point Estimation.

Your Task
Let $X_1, \dots, X_n$ be i.i.d. observations with mean $\mu$ and variance $\sigma^2$. The naive sample variance is $S_n^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})^2$.
(1) Prove that $\mathbb{E}[S_n^2] = \frac{n-1}{n} \sigma^2$.
(2) Conclude that Bessel's correction $S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$ is an unbiased estimator of $\sigma^2$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.