Univariate case

Suppose $X_1,\cdots,X_n \overset{iid}{\sim} N(\mu,\sigma^2)$

Theorem)
① $\overline{X} \sim N(\mu,\frac{\sigma^2}{n})$
② $\overline{X} \perp\!\!\!\perp S, \text{where }S=\frac{1}{n-1}\sum_i (X_i - \overline{X})^2$
③ $(n-1)S/ \sigma^2 \sim \chi^2_{n-1}$
pf)
①

②

③

✔︎ 독립의 불변성

Suppose $X,Y: indep.$
$P(f(X) \in A, g(Y) \in B)=P(X \in f^{-1}(A),Y \in g^{-1}(B))=P(X \in f^{-1}(A))P(Y \in g^{-1}(B))=P(f(X) \in A)P(g(Y) \in B)$

Multivariate case

Suppose $X_1,\cdots,X_n \overset{iid}{\sim} N_p(\mu,\sigma^2)$

① $\overline{X} \sim N_p(\mu, \frac{1}{n}\Sigma)$
② $\overline{X} \perp\!\!\!\perp S \text{ where }S=\frac{1}{n-1}\sum_i (X_i - \overline{X})(X_i - \overline{X})^T$
③ For any $d \neq 0 \in \mathbb{R}^p,(n-1)d^T S d / d^T \Sigma d \sim \chi^2_{n-1}$
④ $(n-1)S \overset{d}{\equiv} \sum_{j=1}^{n-1} Z_j Z_j^T \text{ where } Z_j \overset{iid}{\sim} N_p(o,\Sigma)$
pf)
①

②

③

④