E&R Theorem

Theorem

• For $X_1,\dots ,X_n{\sim }_{iid}N(\mu ,{\sigma }^2)$
• $U$ and $V$ are two different linear combinations of $X_i$s
• $cov[U,V]=0⟺U\perp V$

Linear Combinations

• Say $i=1$
• Then, $U=\overline{x}=\frac{1}{n}{x}_1+\frac{1}{n}{x}_2+\cdots +\frac{1}{n}{x}_n$
• Then, $V=x_i-\overline{x}=x_1-\frac{1}{n}{x}_1+x_2-\frac{1}{n}{x}_2+\cdots +x_n-\frac{1}{n}{x}_n$
• Then, we get $cov[\overline{x},x_i-\overline{x}]$
• $cov[\overline{x},x_i]-cov[\overline{x},\overline{x}]$
• $\frac{1}{n}cov[x_1,x_1]+\frac{1}{n}cov[x_2,x_1]+\cdots -V[\overline{x}]$
• $=\frac{1}{n}V[x_i]-V[\overline{x}]$
• $=\frac{1}{n}{\sigma }^2-\frac{{\sigma }^2}{n}=0$ Therefore, $\overline{X}\perp X_i-\overline{X},\text{ for }i=1,2,\dots ,n$