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**noob mathematician** In order to estimate a population mean, $\displaystyle \mu$, 2 surveys were conducted independently and the statistics were noted. ($\displaystyle \overline{X}_1,\overline{X}_2,\sigma _{\overline{X}_1}, \sigma _{\overline{X}_2} $ are obtained). Assume that $\displaystyle \overline{X_1}$ and $\displaystyle \overline{X_2}$ are unbaised. For some $\displaystyle \alpha$ and $\displaystyle \beta$, the two estimates can be combined to give a better estimator:

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

What choice of $\displaystyle \alpha$ and $\displaystyle \beta$ will minimize the variances, given that $\displaystyle \alpha + \beta =1$?

Since I managed to find one equation which is: $\displaystyle \alpha + \beta =1$. What is another equations which allow me to minimise the variance in order to find suitable $\displaystyle \alpha$ and $\displaystyle \beta$?