If $X_i$ $\sim$ $N(\mu_1, \sigma^2)$ i.i.d, $Y_i$ $\sim$ $N(\mu_2, \sigma^2)$ i.i.d
Prove that $\bar{X}$, $\bar{Y}$, $S_x^2$ and $S_y^2$ are independent. I was told that $\bar{X}$ and $S_x^2$ are independent by Fisher's theorem. Is it possible that after taking a sum of independent random variables - dependence will occur?
Edit: It's known that these two samples are independent
That the sample mean and the sample variance of a sample of i.i.d. Normals are independent Statistics, can be derived as a consequence of Cochran's Theorem, and it is one of the characterizations of the Normal distribution (it holds only for the Normality case).
In your case, we are not told what is the relation between the sample from $X$ and the sample from $Y$.