Let $X_1, X_2, \dots, X_n$ be i.i.d with $E[X_i]=\mu$, Var$(X_i)= \sigma^2$. random variables. Define the sample mean as $\overline{X} = \frac{1}{n} \sum_{i=1}^{n} X_i$.
My question is how to intuitively understand the independence of $X_1 - \overline{X}$ and $X_2 - \overline{X}$; and the independence of $\overline{X}$ and $X_i - \overline{X}$.
Here is my thoughts:
$X_1-\overline{X}$ and $X_2-\overline{X}$ are independent, because $\overline{X}$ is like a fixed value in this case (please correct me if I'm wrong).
For the second question, I've calculated the covariance of $\overline{X}$ and $X_i - \overline{X}$, which equals to 0, so they are supposed to be independent. But I'm confused about how to understand their independence intuitively. From my understanding, the change of $\overline{X}$ will change the value of $X_i - \overline{X}$, so they are not independent.
This confuses me so much, any insight will be appreciated!
$X_1-\bar X$ and $X_2-\bar X$ are not independent except in trivial cases. For $n=2$ you have $(X_1-\bar X)=−(X_2-\bar X)$.
If $X_1,X_2$ are iid $\pm1$ with equal probability, then $X_i-\bar X=0\iff \bar X \in \{\pm1\}$ and $X_i-\bar X\in \{\pm1\}\iff \bar X=0 $, so in that non-Gaussian case $X_i-\bar X$ is not independent of $\bar X$ despite the zero covariance.
If the $X_i$ are iid Gaussian then $(X_i-\bar X,\bar X)$ is bivariate Gaussian and there is zero covariance between them, so $X_i-\bar X$ and $\bar X$ are independent.