Let $x_1,x_2,\ldots,x_n$ and $y_1,y_2,\ldots,y_n$ be uniformly, independent random variables. What can I say about the distribution of the following two variables?
$$S_1 = (x_1,x_2,\ldots,x_n,-\sum(x_i))$$
$$S_2 = (y_1,y_2,\ldots,y_n,-\sum(y_i))$$
My guess is that they have exactly the same distribution, but got stuck on how to argue for it. It would be easier without $\sum(x_i)$ and $\sum(y_i)$, since both $S_1$ and $S_2$ are made of independent identical random variables.