Two surveys were independently conducted to estimate a population mean, $\displaystyle \mu$. Denote the estimates and their standard errors by $\displaystyle \overline{X}_1$ and $\displaystyle \overline{X}_2$ and $\displaystyle \sigma _{\overline{X}_1}$ and $\displaystyle \sigma _{\overline{X}_2}$. Assume that $\displaystyle \overline{X}_1$ and $\displaystyle \overline{X}_2$ are unbiased. For some $\displaystyle \alpha$ and $\displaystyle \beta$, the two estimates can be combined to give a better estimator:

$\displaystyle X=\alpha \overline{X}_1 + \beta \overline{X}_2$

a.Find the conditions on $\displaystyle \alpha$ and $\displaystyle \beta$ that make the combined estimate unbiased.

b.What choice of $\displaystyle \alpha$ and $\displaystyle \beta$ minimizes the variances, subject to the condition of unbiasedness?

Answer: (a) $\displaystyle \alpha + \beta =1$

Answer: (b) $\displaystyle \alpha= \frac{\sigma^2 _{\overline{X}_2}}{\sigma^2 _{\overline{X}_2}+\sigma^2 _{\overline{X}_1}}$, $\displaystyle \beta= \frac{\sigma^2 _{\overline{X}_1}}{\sigma^2 _{\overline{X}_2}+\sigma^2 _{\overline{X}_1}}$

I don't know how to get the answer for part (b)! please help