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If $(x_1,x_2,\ldots,x_n)$ be a random sample drawn from a normal population with mean $\mu$ and standard deviation $\sigma,$ then find the sampling of $T=\sqrt{\sum_{i=1}^n (x_i-\bar{x})^2}.$ Further show that $\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i$  and $T$ are independently distributed.

I arrived to the fact that $\frac{ns^2}{\sigma^2}$ is a chi square with $(n-1)$ d.f. but don't know how to proceed here.

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    $\begingroup$ What is $s^2$? I did not find you defined it. $\endgroup$
    – user158565
    Commented Jun 11, 2019 at 19:31
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    $\begingroup$ You could use several of the answers at stats.stackexchange.com/questions/151096 along with the (algebraic) fact that $T$ is a function of the differences $x_i-x_j.$ $\endgroup$
    – whuber ♦
    Commented Jun 11, 2019 at 19:47
  • $\begingroup$ will you please elaborate more?Here, $ns^2=\sum_{i=1}^{n} (x_i-\bar{x})^2$ $\endgroup$
    – who
    Commented Jun 12, 2019 at 3:03

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