Covariance of estimated population totals in a cluster sample. Suppose a one-stage cluster sample is taken from a population of N psus, with inclusion probabilities πi.
Let tx and ty be the population totals for response variables x and y, and let tix and tiy be the totals of variables x and y in psu i.
a. Show that

b. If an SRS of n of the N psus is selected, with πi = n/N and πik = (n/N) [ (n−1) / (N − 1)], show using part (a) that