I have 2 independent random samples: {$X_1, X_2$} and {$Y_1, Y_2...Y_{22}$} with E[$X_i$] = $\mu$ and E[$Y_j$] = $3\mu$, Var[$X_i$]= $\sigma^2$ and Var[$Y_j$]= $\sigma^2/2$.
Now I have the following unbiased estimator ->
$T_1$(X,Y) = $\frac{\bar X+\bar Y}{4}$.
I know that since it is unbiased it's MSE is just it's variance. I'm struggling to find the variance.
For the variance of $T_1(X,Y)$ we have, \begin{align} \text{Var}\left(\frac{\overline{X}+\overline{Y}}{4}\right) &= \frac{1}{16}\text{Var}(\overline{X}+\overline{Y}) = \frac{1}{16}\left(\text{Var}(\overline{X})+\text{Var}(\overline{Y}) + 2\text{Cov}(\overline{X},\overline{Y})\right) \\ &= \frac{1}{16}\left(\text{Var}(\overline{X})+\text{Var}(\overline{Y})\right) \end{align}
Since, the two are independent. Then for the variances of the sample means we have, \begin{align} \text{Var}(\overline{X}) = \frac{1}{2}\text{Var}(X) = \frac{\sigma^2}{2}\\ \text{Var}(\overline{Y}) = \frac{1}{22}\text{Var}(Y) = \frac{\sigma^2}{44} \\ \end{align}
Hope this helps.