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
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

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

1 , . . . ,τ

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?

Why Choose Us

  • 100% non-plagiarized Papers
  • 24/7 /365 Service Available
  • Affordable Prices
  • Any Paper, Urgency, and Subject
  • Will complete your papers in 6 hours
  • On-time Delivery
  • Money-back and Privacy guarantees
  • Unlimited Amendments upon request
  • Satisfaction guarantee

How it Works

  • Click on the “Place Order” tab at the top menu or “Order Now” icon at the bottom and a new page will appear with an order form to be filled.
  • Fill in your paper’s requirements in the "PAPER DETAILS" section.
  • Fill in your paper’s academic level, deadline, and the required number of pages from the drop-down menus.
  • Click “CREATE ACCOUNT & SIGN IN” to enter your registration details and get an account with us for record-keeping and then, click on “PROCEED TO CHECKOUT” at the bottom of the page.
  • From there, the payment sections will show, follow the guided payment process and your order will be available for our writing team to work on it.