probabilistic mathematical statistics problem solving.

Because two quantities Xi and Xj which are independent of each other have Cov(Xi,Xj)=0

So Cov(Xi,X both)=Cov(Xi,(1/n)(X1+X2+... +Xn))

=Cov(Xi,x1/n)+Cov(Xi,x2/n)+... +Cov(Xi,Xi/n)+... +Cov(Xi,Xn/n)

=Cov(Xi,Xi/n)

=(1/n)Cov(xi,xi)

=(1/n)D(xi)