Question: Given n ≥ 2 numbers (a1, a2,...,an) with no repetitions, a bootstrap sample is a sequence (x1, x2,...,xn) formed from the aj 's by sampling with replacement with equal probabilities. Bootstrap samples arise in a widely used statistical method known as the bootstrap. For example, if n = 2 and (a1, a2) = (3, 1), then the possible bootstrap samples are (3, 3),(3, 1),(1, 3), and (1, 1).
(a) How many possible bootstrap samples are there for (a1,...,an)?
(b) How many possible bootstrap samples are there for (a1,...,an), if order does not matter (in the sense that it only matters how many times each aj was chosen, not the order in which they were chosen)?
(c) One random bootstrap sample is chosen (by sampling from a1,...,an with replacement, as described above). Show that not all unordered bootstrap samples (in the sense of (b)) are equally likely. Find an unordered bootstrap sample b1 that is as likely as possible, and an unordered bootstrap sample b2 that is as unlikely as possible. Let p1 be the probability of getting b1 and p2 be the probability of getting b2 (so pi is the probability of getting the specific unordered bootstrap sample bi). What is p1/p2? What is the ratio of the probability of getting an unordered bootstrap sample whose probability is p1 to the probability of getting an unordered sample whose probability is p2?