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In: Statistics and Probability

Problem 5.10.10 Suppose you have n suitcases and suitcase i holds Xi dollars where X1, X2,...

  1. Problem 5.10.10

Suppose you have n suitcases and suitcase i holds Xi dollars where X1, X2, …, Xn are iid continuous uniform (0, m) random variables. (Think of a number like one million for the symbol m.) Unfortunately, you don’t know Xi until you open suitcase i.

            Suppose you can open the suitcases one by one, starting with suitcase n and going down to suitcase 1. After opening suitcase i, you can either accept or reject Xi dollars. If you accept suitcase i, the game ends. If you reject, then you get to choose only from the still unopened suitcases.

            What should you do? Perhaps it is not so obvious? In fact, you can decide before the game on a policy, a set of rules to follow. We will specify a policy by a vector (t1, … tn) of threshold parameters.

  • After opening suitcase i, you accept the amount Xi if Xiti .
  • Otherwise, you reject suitcase i and open suitcase i-1.
  • If you have rejected suitcases n down through 2, then you must accept the amount X1 in suitcase 1. Thus the threshold t1=0 since you never reject the amount in the last suitcase.
  1. Suppose you reject suitcase n through i+1, but then you accept suitcase i. Find E[Xi|Xiti ].
  2. Let Wk denote your reward given that there are k unopened suitcases remaining. What is E[W1]?
  3. As a function of tk, find a recursive relationship for E[Wk] in terms of tk and E[Wk-1].
  4. For n=4 suitcases, find the policy (t1*, … t4*) that maximizes E[W4].

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