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Show that for independent random variables $X_{1}, \ldots, X_{n}$, the random variables $$ \bar{X}=\frac{1}{n} \sum \limits_{i=1}^{n} X_{i}, \quad S^{2}=\frac{1}{n-1} \sum \limits_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2} $$ are independent.

Idea: I have found the following similar lemma, but it has a rather complicated proof. Unfortunately, so far I have not come across an easier way to show the above statement. Can someone help me further?

Lemma 7.4.9. If $ X \sim N\left(\mu, \sigma^{2}\right), X_{1}, \ldots, X_{n} $ are independent identically distributed random variables, $ X_{i} \stackrel{d}{=} X $, then $ \bar{X}_{n} $ and $ S_{n}^{2} $ are independent.

clementine1001
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    Isn't the standard proof to use Basu's theorem on complete and ancillary statistics? Also, I'm pretty sure the claim is true if and only if your data is gaussian. – Andrew Jul 14 '23 at 18:15
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    @AndrewZhang See e.g. this – Robert Israel Jul 14 '23 at 18:29
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    @RobertIsrael Proposition 1 is in your linked document is exactly what AndrewZhang is saying... – gandalfbalrogslayer Jul 14 '23 at 18:30
  • Yes, that's why I posted the link. – Robert Israel Jul 14 '23 at 18:33
  • Thanks to all of you for providing the appropriate theorem with proof or derivation here. Now I am actually even more puzzled by the task, since one had to show exactly, as I wrote it above, in an exam with only 90 minutes... – clementine1001 Jul 15 '23 at 19:02
  • @clementine1001 Basu's theorem is not needed and it is arguably not a standard proof of this result. A standard proof is a change of variables: https://math.stackexchange.com/q/2952346/321264. – StubbornAtom Jul 16 '23 at 09:42

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