= Independence of the normal sample mean and sample variance
{title2=$\overline X\perp S^2$}
For an independent <normal sample> with common mean $\mu$ and positive <variance> $\sigma^2$, its <sample mean> and <sample variance> are independent. Indeed, the mean and residual vector $(X_i-\overline X)_i$ are jointly <multivariate normal>, and $\operatorname{Cov}(\overline X,X_i-\overline X)=\sigma^2/n-\sigma^2/n=0$. Vanishing cross-covariance between jointly Gaussian blocks proves independence of the mean from the entire residual vector. Since the <sample variance> is a measurable <function> of that vector, independence follows. Independence from each coordinate separately would not suffice without the joint Gaussian argument.
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