please help with question 1

Let Y1,Y2,,Yn represent the systolic blood pressure measurements of a random sample of n patients aged x1,x2,,xn, respectively. Suppose that Yi=Xi+i,i=1,2,,n where x1,x2,,xn are the exactly known ages of the patients, is an unknown parameter to be estimated fror the data, and 1,2,,n are independent errors with E(i)=0andV(i)=2,i=1,2,,n. (a) Calculate E(Yi) and V(Yi). (b) Consider =i=1nxii=1nYi as an estimator of . (i) Show that E()=. (ii) Show that ~ is a consistent estimator of if limn(i=1nxi)2n=0. (c) Consider ~=n1i=1nxiYi as an estimator of . (i) Show that E(~)=. (ii) Show that ~ is a consistent estimator of if limnn21i=1nxi21=0. (d) Consider =i=1nxi2i=1nYixi as an estimator of . (i) Show that E(^)=. (ii) Show that is a consistent estimator of if limni=1nxi21=0. Let Y1,Y2,,Yn represent the systolic blood pressure measurements of a random sample of n patients aged x1,x2,,xn, respectively. Suppose that Yi=Xi+i,i=1,2,,n where x1,x2,,xn are the exactly known ages of the patients, is an unknown parameter to be estimated fror the data, and 1,2,,n are independent errors with E(i)=0andV(i)=2,i=1,2,,n. (a) Calculate E(Yi) and V(Yi). (b) Consider =i=1nxii=1nYi as an estimator of . (i) Show that E()=. (ii) Show that ~ is a consistent estimator of if limn(i=1nxi)2n=0. (c) Consider ~=n1i=1nxiYi as an estimator of . (i) Show that E(~)=. (ii) Show that ~ is a consistent estimator of if limnn21i=1nxi21=0. (d) Consider =i=1nxi2i=1nYixi as an estimator of . (i) Show that E(^)=. (ii) Show that is a consistent estimator of if limni=1nxi21=0