Prove $\bar{X}$ and $X_i-\bar{X}$ are independent if $X_i's$ are independently normally distributed The statement comes from the book statistics and data analysis page 179, and I just wonder how to prove that $\bar{X}$ and $X_i-\bar{X}$ are independent if $\require{enclose}\enclose{horizontalstrike}{Cov(\bar{X},X_i-\bar{X})=0}$ and $X_i's$ are independently normally distributed.

(Note: In this answer I'll only consider standard normal $X_i$s, since it's no harder to do when they have other means and variances)
One way of doing this is with moment generating functions.
For a bivariate random variable $Z = (Z_1, Z_2)$, $Z_1$ and $Z_2$ are independent if and only if the moment generating function $M_Z(\mathbf{s}) = \mathbf{E} e^{\mathbf{s}^T Z}$ factors into a product of a functions of $s_1$ and $s_2$ only, where $\mathbf{s} = (s_1, s_2)$.
Now, the thing to notice is that $(\bar{X}, X_i - \bar{X})$ is a linear transformation of the whole sample $X = (X_1, \dotsc, X_n)$.
Namely,
$$
\begin{pmatrix}\bar{X} \\ X_i - \bar{X}\end{pmatrix}
= \begin{pmatrix}\mathbf{1}^T/n \\ e_i^T - \mathbf{1}^T/n\end{pmatrix} X
:= H X
$$
where $\mathbf{1}$ is a vector of all $1$s and $e_i$ is the vector of all $0$s with a $1$ in the $i$th entry.
We can use this to write down the MGF:
\begin{align*}
\mathbf{E} \exp\biggl\{\mathbf{s}^T \begin{pmatrix}\bar{X} \\ X_i - \bar{X}\end{pmatrix}\biggr\}
= \mathbf{E} \exp\{\mathbf{s}^T H X\} 
= \mathbf{E} \exp\{(H^T\mathbf{s})^T X\} 
\end{align*}
Then we can use the known moment generating function of a vector of iid normals ($\mathbf{E} e^{\mathbf{t}^T X} = \exp\{\frac{1}{2}\mathbf{t}^T \mathbf{t}\}$) to conclude that
$$
\mathbf{E} \exp\{(H^T\mathbf{s})^T X\} 
= \exp\{\frac{1}{2}(H^T\mathbf{s})^T(H^T\mathbf{s})\}
= \exp\{\frac{1}{2}\mathbf{s}^T HH^T \mathbf{s}\}.
$$
Now, we can multiply
\begin{align*}
HH^T
&= \begin{pmatrix}\mathbf{1}^T/n \\ e_i^T - \mathbf{1}^T/n\end{pmatrix} \begin{pmatrix}\mathbf{1}/n , & e_i - \mathbf{1}/n\end{pmatrix} \\
&= \begin{pmatrix}
\mathbf{1}^T/n \mathbf{1}/n & \mathbf{1}^T/n (e_i - \mathbf{1}/n) \\
(e_i - \mathbf{1}/n)^T \mathbf{1}/n & (e_i - \mathbf{1}/n)^T(e_i - \mathbf{1}/n)
\end{pmatrix} \\
&= \begin{pmatrix}
1/n & 0 \\
0 & 1 - 1/n
\end{pmatrix},
\end{align*}
so that the MGF becomes
$$
\exp\biggl\{\frac{1}{2}\mathbf{s}^T HH^T \mathbf{s}\biggr\}
= \exp\biggl\{\frac{1}{2}\biggl(\frac{1}{n}s_1^2 + \frac{n-1}{n}s_2^2\biggr)\biggr\}
= e^{\frac{1}{2n}s_1^2} e^{\frac{n-1}{2n} s_2^2}.
$$
Since this factors into terms containing only $s_1$ and $s_2$ respectively, we conlcude that $\bar{X}$ and $X_i - \bar{X}$ are independent (and also that they have $N(0, 1/n)$ and $N(0, 1 - 1/n)$ distributions respectively).
