(Survey sample) In a kingdom of N total citizens, 3 people, Alice, Bob and Cathy, run for the president. Some gambling company did a preliminary survey sample of size n on the support rate with simple random sampling with replacement. This means they draw X1,X2,Xn i.i.d sample that satisfy: P(Xi=xj)=N1, while i=1,2,n stands for the n draws, and xj {(1,0,0),(0,1,0),(0,0,1)},j=1,2,N stands for who the j-th citizen supports. Here xj=(1,0,0) mears the j-th citizen supports Alice, (0,1,0) stands for Bob, and (0,0,1) stands for Cathy. (a) (10') Show that Xi follows the same distribution as Y where Y follows multinomial distribution Multi (1,p) where =(1,2,3)= (NN1,NN2,NN2) are the actual support rate of Alice, Bob and Cathy. Here Ni(i=1,2,3) is the actually number of citivens that support Alice, Bob and Cathy respectively. Remark: rememher when Y=(Y1,Y2,Y3) follows multi (1,~) it means one of Y1,Y2,Y3 is 1 and other two are 0 and P(Yi=1)=i. (b) (10") What is the covariance matrix for Y ? (Hint: it should be a 33 matrix.)