I got a problem as follows:

If $\overline X_1$ is the mean of a random sample of size $n$ from a normal distribution with the mean $\mu$ and the variance $\sigma_{1}^2$, $\overline X_2$ is the mean of a random sample of size $n$ from a normal distribution with the mean $\mu$ and the variance $\sigma_{2}^2$, and the two samples are independent, show that:
(a)$\ \omega \overline X_1+(1-\omega)\overline X_2$, where $0\le \omega \le 1$, is an unbiased estimator of $\mu$;
(b) the variance of this estimator is a minimum when $\omega =\frac{\sigma_{2}^2}{\sigma_{1}^2+\sigma_{2}^2}$.

Now, I have done part (a), but have trouble in part (b) since I have no idea about dealing with two variables and how I could use the UMVUE equality.

Any help on part (b) please.

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Hints:

1- How do you calculate the variance of a random variable (estimator)?

2- How do you find the maximum or a minimum of a function based on some parameters; $df(x)/dx=0$?

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Well, my biggest concern about this problem is what my density function f(x) is since this is a mixed distribution. I just did not figure that out. Do you have a hint on the f(x)? – Scorpio19891119 Jan 28 '13 at 4:00
I wrote $f(x)$ as just some function. it is not density.. Consider $E[X^2]$ and that expectation is a linear operation.. Now you will replace $X^2$ with your estimator right? – Seyhmus Güngören Jan 28 '13 at 9:44