QUES IION 1 $|18|$ (a) Prove that if boin $T_{1}$ and $T_{2}$ dre unbiased estimators of $\theta$, then $c T_{1}+(1-0) T_{2}$ is also an unbidsed estimator of $\theta$ tor any real valued $c \in[0,1]$ (b) If $T_{1}$ and $T_{2}$ are indupendent unbiased estimators of $\theta$ such that $V\left(7_{1} ight)=\tau^{2}$ and $V\left(T_{2} ight)=2 \tau^{2}$, find $c$ which minimizes the variance of the estimator $T=c T_{1}+(1-c) L_{2}$ of $\theta$ (c) Consider the estimators $7_{1}, L_{2}$, and $T$ (cvaluated at $c$ found 11 part (b)) of $0 \mathrm{n}$ part (b) Determine $e f f\left(7_{2}, 7_{1} ight) $ and $e f f\left(T_{2}, T ight) $ Which among the three estimators is the best estimator of $\theta$ and why'? SP.PB.088