Question $12$:

Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent, identically distributed random variables from a population with mean $$ and variance ${}^2.$ Let $\frac{?}{\text{b}\text{a}\text{r}}(Y)=\frac{1}{4}(Y_1+Y_2+Y_3+Y_4)$ denote the average of these four random variables.

What are the expected value and variance of $\frac{?}{\text{b}\text{a}\text{r}}(Y)$ in terms of $$ and ${}^2?$

Now, consider a different estimator of $$ :

$W=\frac{1}{8}{Y}_1+\frac{1}{8}{Y}_2+\frac{1}{4}{Y}_3+\frac{1}{2}{Y}_4$

This is $\text{'}$an example of a weighted average of the $Y_i.$ Show that $W$ is also an unbiased estimator of $.$ Find the variance of $W.$

$3.$ Based on your answers above, which estimator of $$ do you prefer, $\frac{?}{\text{b}\text{a}\text{r}}(Y)$ or $W?$