(Hard.) There is a population of N subjects, indexed by i = 1,...,N.

Associated with subject i there is a number vi. A sample of size n is chosen at random without replacement.

(a) Show that the sample average of the v’s is an unbiased estimate of the population average. (There are hints below.)

(b) If the sample v’s are denoted V1, V2,...,Vn, show that the probability distribution of V2, V1,...,Vn is the same as the probability distribution of V1, V2,...,Vn. In fact, the probability distribution of any permutation of the V ’s is the same as any other: the sample is exchangeable.

Hints. If you’re starting from scratch, it might be easier to do part (b)

first. For (b), a permutation π of {1,...,N} is a 1–1 mapping of this set onto itself. There are N! permutations. You can choose a sample of size n by choosing π at random, and taking the subjects with index numbers

π(1), . . . , π(n) as the sample.