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$X_1, X_2, \cdots, X_n$ are independent Gaussian $N(0,1)$ random variables. I need to show that the following random variables in $\mathbb{R}^2$ have the same distribution:
$\displaystyle U=\left(\bar{X}, \sum_{k=1}^n(X_k-\bar{X})^2\right)$ and $\displaystyle V=\left(n^{-\frac{1}{2}}X_n, \sum_{k=1}^{n-1}X_k^2 \right)$. Here $\bar{X} $ denotes the average of $X_1, \cdots, X_n$.

So far I can see that the components of $V$ are independent so $P(V_1\le x, V_2\le y)=P(V_1\le x)P(V_2\le y)$. I can also see that $U_1$ and $V_1$ have the same distribution (which they must). I don't directly see how $U_2$ and $V_2$ have the same distribution, although they must if $U$ and $V$ do.

Assuming the conclusion, the components of $U$ and $V$ must be identically distributed, respectively. Then the independence of $V_1$ and $V_2$ implies the independence of $U_1$ and $U_2$. Therefore the work reduces to showing that $u_1$ and $U_2$ are independent and $U_2$ and $V_2$ have the same distribution.

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  • $\begingroup$ You must exactly show this: that $U_1$ and $U_2$ are independent (they are) and that $U_2$ and $V_2$ are identically distributed. $\endgroup$ – Did Jan 16 '13 at 14:28
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    $\begingroup$ math.stackexchange.com/q/47350/48763 $\endgroup$ – Learner Jan 16 '13 at 14:30
  • $\begingroup$ Yes, I'll edit my post. $\endgroup$ – Spook Jan 16 '13 at 14:30

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