Optimizing Estimator Variance with Cramer-Rao Inequality

[EnzoF61]

If $$\overline{_}X1$$ and $$\overline{_}X2$$ are the means of independent random samples of size n1 and n2 from a normal population with the mean $$\mu$$ and $$\sigma^2$$, show that the variance of the unbiased estimator Var($$\omega$$$$\overline{_}X1$$ +(1-$$\omega$$)$$\overline{_}X2$$) is a minimum when $$\omega$$= n1 / (n1 + n2).

My professor gave a hint to find the Var($$\omega$$$$\overline{_}X1$$ +(1-$$\omega$$)$$\overline{_}X2$$) and then to minimize with respect to $$\omega$$ using first and second derivatives.

I understand to square out the coefficients due to independence but I'm not sure where to begin to find the variances of the means of independent random samples. I feel I should have an understanding to minimize with the Cramer-Rao Inequality. Maybe I'm trying to look too deep into what is being asked?

 

[Mandark]

Can you show that $$\mbox{Var}(\overline{X_1}) = \frac{\sigma^2}{n_1}$$? Hint: write $$\overline{X_1}$$ as a sum of independent and identically distributed RVs and use $$\mbox{Var}(\sum Y_i) = \sum \mbox{Var}(Y_1)$$ as well as pulling out a constant becomes multiplying by the square of the constant.