Let $X_1,\ldots,X_n$ be a random sample from $N(\mu, \sigma^2)$. Show that the sample mean and each $X_i-\bar X, i= 1,\ldots,n$, are iid. Actually $\bar X$ and the vector $(X_1-\bar X,\ldots,X_n-\bar X)$ are independent and this implies that $\bar X$ and the summation of the vector squared are independent. Thus we could find the joint distribution of $\bar X$ and $nS^2/\sigma^2$ using this result.
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1 Answer
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Note: $\bar{X}$ and each $X_i-\bar{X}$ are not identically distributed: for instance, $\mathbb E[\bar{X}] \ne \mathbb E[X_i-\bar{X}]$, for non-zero mean $\mu$. However, to prove that $\bar{X}$ is independent of each $X_i-\bar{X}$, one can proceed using the following hints.
Hints:
- Jointly normally distributed random variables that are uncorrelated are indepenedent see here;
- Argue that $\bar{X}$ and the $X_i-\bar{X}$ are jointly normal by using the definition of joint normality and the fact that the $X_i$ are independent identically distributed samples from a Gaussian distribution;
- Show that each pair of variables $\left\{\bar{X}, X_i-\bar{X}\right\}$ is uncorrelated.
