For $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$ iid, the sample mean $\bar{X}$ and the sample variance $S^2$ are independent. Which justification is cleanest and most general?

Basu's Theorem

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For X_{1}, \dots, X_{n} \sim N (μ, σ^{2}) iid, the sample mean \overset{ˉ}{X} and the sample variance S^{2} are independent. Which justification is cleanest and most general?