Show that n

∑

i,j=1 Y

r i Y s

j = nk rs + n (n − 1)k

(r)(s)

.

Hence deduce that, under simple random sampling, the average value over all samples of n −2∑n i,j=1 Y r i Y s j is n −1Krs + (1 − n −1)K(r)(s) = n −1Kr,s + K(r)(s)
while the same function calculated in the population is N −2 N ∑
i,j=1 Y r i Y s j = N −1Kr,s + K(r)(s)
.