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Theorem: Suppose that $Y_1, \dots Y_n \sim \mathcal{N}(\mu, \sigma^2)$ and these variables are mutually independent. Then $\overline{Y} = \frac{1}{n} \sum_{i=1}^n Y_i$ and $S^2 = \frac{1}{n-1}\sum_{i=1}^n (Y_i - \overline{Y})^2$ are independent variables.

I have a problem with the proof of this theorem and I will only post the relevant sections.

Proof: Define for $j= 1, \dots, n$: $X_j = \sigma Z_j + \mu$

where $Z_1, \dots , Z_n$ are independent variables that are standard normally distributed. I.e., $Z_i \sim \mathcal{N}(0,1)$.

My book then provides argumentation that $\overline{X}$ and $S_X^2$ (defined similarly as the things written in the theorem) are independent variables.

How is this sufficient to conclude that $\overline{Y}$ and $S_{Y}^2$ are independent?

Note that $X_j \sim Y_j$, and this should probably be used but I can't see how.

Thanks in advance.

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The "theorem" as-quoted does not hold, as the following counterexample shows:

With $n=3$ and $Y_1\sim\mathcal{N}(0,1)$, suppose $Y_2=Y_1$ and $Y_3=-Y_1$. Now we have $Y_1,Y_2,Y_3\sim\mathcal{N}(0,1),$ but $$\begin{align}\overline{Y}&={Y_1+Y_2+Y_3\over 3}={1\over 3}Y_1\\[2ex] S^2&={1\over 2}\left((Y_1-{1\over 3}Y_1)^2+(Y_2-{1\over 3}Y_1)^2+(Y_3-{1\over 3}Y_1)^2\right)={4\over 3}Y_1^2 \end{align}$$ so $S^2=12\,(\overline{Y})^2$; hence, $\overline{Y}$ and $S^2$ are not independent.

Perhaps you've omitted to quote some part of the context where the author states an assumption that the $Y_i$ are mutually independent (in which case the theorem is valid)? Would you mind citing the book and what page this is on?

EDIT:

Given the question's revision to have the $Y_i$ mutually independent ...

From what you've quoted, it isn't clear what kind of "independence" is being assumed for the $X_j$, but for Normal random variables, pairwise independence implies mutual independence, so the theorem holds in either case. Thus, writing $X_j = \sigma Z_j + \mu$, where the $Z_i$ are (pairwise) independent standard Normal, is equivalent to saying $X_1, \dots X_n \sim \mathcal{N}(\mu, \sigma^2)$ where the $X_j$ are mutually independent. So, whatever is proven about the distribution of the $X_j$, the same will hold for the distribution of the $Y_i$.

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  • $\begingroup$ Ah sorry yes that was an assumption the book made. Edited the post $\endgroup$ – user370967 Apr 25 '18 at 6:36
  • $\begingroup$ As for the book, it is a small textbook printed in dutch made by my uni professor and not for sale. $\endgroup$ – user370967 Apr 25 '18 at 6:39

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