We consider a population of size N = 3, such that U = {1, 2, 3}. The map of following poll is adopted: p({1, 2}) = 1/2 ; p({1, 3}) = 1/4 ; p({2, 3}) = 1/4 .

We invite you to answer the following questions.

a) Find the first order inclusion probabilities.

b) Find the quantities kl.

c) Find the probability distribution of the Horvitz-Thompson estimator.

d) Find the probability distributions of the two variance estimators seen in class. What are the biases of these estimators?

e) Find the probability distributions of the two variance estimators in the special case yk = k, k U. Discuss.