distribution of $\sum_{i=1}^n (X_i-X_{n+i})^2$ where $X_1,X_2,\dots,X_{2n}$ are iid $N(\mu,\sigma^2)$

Question

Suppose $X_1,X_2,\dots,X_{2n}$ are iid $\mathcal N(\mu,\sigma^2)$ random variables.

How can I find the distribution of $\displaystyle\sum_{i=1}^n (X_i-X_{n+i})^2$?
Should I approach it by taking $Z=\dfrac{X-\mu}{\sigma}$ where $Z\sim \mathcal N(0,1)$?

Any hint will also help me.

Answer 1

Note that $X_i-X_{n+i}$ is an independent sequence. Further note that $$X_i-X_{n+i} \sim \mathcal{N}(0,2\sigma^2),$$ hence $$\left(\frac{X_i-X_{n+i}}{\sqrt2\sigma}\right)^2\sim\chi^2_{(1)},$$which is also an independent sequence. Therefore $$\frac{1}{2\sigma^2}\sum_{i=1}^n (X_i-X_{n+i})^2\sim\chi^2_{(n)}.$$ Now use this page to see how a scaled chi-dist is related to a Gamma distribution.