There are a lot of symbols so I am sorry if this is hard to read.
Q=theta hat, q=theta
Suppose E(Q1)=E(Q2)=q. V(Q1)=(sigma1)^2, V(Q2)=(sigma2)^2. Consider the estimator Q3=aQ1+(1-a)Q2.
a)Show that Q3 is an unbiased estimator for q.
b) If Q1 and Q2 are independent, how should the constant a be chosen in order to minimize the variance of Q3?