Let $X_1,...,X_m$ be i.i.d. sample with $N(\mu_1,\sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(\mu_2,2\sigma^2)$.
Let $S_x^2 = \sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= \sum_{i=1}^n(Y_i− Y_n)^2$.
(b) Determine the values of $\alpha$ and $\beta$ for which $\alpha S_x^2 + \beta S_y^2$ will be an unbiased estimator with minimum variance.
It is not so difficult to solve if $S_x^2 = \sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= \sum_{i=1}^n(Y_i− Y_n)^2$ are independent. However I don't know how to show that they are independent... any possible helps??