finish the questions in the attched files , due in 48 hours

16:332:541 Stochastic Signals and Systems

Assignment 1

Due: October 8, 2021

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 about whether to accept suitcase the Xi dollars in

suitcase i. We will specify a policy by a vector (τ1, . . . ,τn) of threshold parameters such that

• After opening suitcase i, you accept the amount Xi if Xi ≥ τi; otherwise, you reject

suitcase i and open suitcase i− 1.

Your would like to choose a policy (τ∗1 ,τ

∗

2 , . . . ,τ

∗

n) that maximizes your expected winnings.

(a) Find the conditional expected value E[Xi|Xi ≥ τi].

(b) Let W1(τ1) denote your winnings given there is just 1 unopened suitcase and your

threshold is τ1. What is E[W1(τ1)] as a function of τ1? We can write the value of τ1

that maximizes E[W1(τ1)] as

τ∗1 = arg max

τ1

E[W1(τ1)].

Show that τ∗1 = 0. That is, you should never reject what you get in the last suitcase.

(c) Let Wk(τ1, . . . ,τk) denote your reward given that there are k unopened suitcases re-

maining and the thresholds are τ1, . . . ,τk. As a function of τk, find a recursive rela-

tionship for E[Wk(τ1, . . . ,τk)] in terms of τk and E[Wk−1(τ1, . . . ,τk−1)].

(d) When there are 2 suitcases left, you choose the threshold

τ∗2 = arg max

τ2

E[W2(τ2,τ

∗

1 )]

What is τ∗2 ? (Keep in mind that τ

∗

1 = 0.)

(e) Suppose you have already found the optimized thresholds τ∗1 , . . . ,τ

∗

k−1. When there are

k suitcases left, you choose the optimized threshold

τ∗k = arg max

τk

E

[

Wk(τ

∗

1 , . . . ,τ

∗

k−1,τk)

]

Find the policy the optimized policy (τ∗1 , . . . ,τ

∗

4 ).

(f) Define W∗k = Wk(τ

∗

1 , . . . ,τ

∗

k ) and αk = E[W

∗

k ]/m. How are α1,α2, . . . ,αk related?

(g) What is limk→∞ αk?

1

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