Overview
DO NOT USE CHAT GPT. ANSWERS ARE CHAT GPT PROOF. *IF R IS USED PLEASE PROVIDE CODE.
Question 11 pts
A decision-maker is willing to pay $50 for a prospect that gives a 60% chance of receiving $100 and a 40% chance of receiving nothing. Her willingness to pay does not depend on her initial level of wealth.
She is offered the right to sell each item in a collection of 100 pieces of artwork. The sale price
of each piece is an independent draw from a normal distribution with a mean of $300 and a standard deviation of $50.
What is the maximum amount the decision-maker should be willing to pay for the right to sell all 100 pieces of artwork?
(Please round your answer to the nearest whole dollar, like 1234.)
Flag question: Question 2Question 21 pts
Sam has a 2:00 PM flight out of Denver International Airport. (The flight boards at 2:00 and departs at 2:30, but we will only look at what happens before Sam boards the flight.) Travel time to the airport is normally distributed with mean 60 minutes and standard deviation 10 minutes. Time to get to the gate once he has reached the airport is normally distributed with a mean of 30 minutes and a standard deviation of 5 minutes. If he is not at the gate by 2:00 PM he misses flight, but he can catch another one at 4:00 at no extra fee, but this increases his time at the airport by 2 hours. Sam values his time at the airport at -$40/hr. – the less time he spends there, the better.
- To minimize the expected loss from his time spent at the airport (loss= $40/hr. x hours spent at the airport), how many minutes before 2:00 should Sam leave for the airport? Please round your answer to the nearest whole minute, like 123.
- If Sam values his time at the airport at $10/hr., then how many minutes before 2:00 should Sam leave for the airport?
Please round your answer to the nearest whole minute, like 123.
Flag question: Question 3Question 31 pts
Joe is scheduled to meet a friend for lunch at a restaurant at 12:00. Joe aims to get there at 12:00, but he might be early or late. His time to get there has an exponential distribution with a mean of 25 minutes. He cares equally about being early or late: his loss is the number of minutes before or after noon that he arrives. How many minutes before noon should he leave to minimize the expected loss from being early or late? Please round your answer to the nearest whole minute, like 12.
Flag question: Question 4Question 41 pts
Suppose that in Problem 3, Joe’s friend is also trying to arrive at 12:00 and that his time to get there also has an exponential distribution with a mean of 25 minutes. However instead of caring equally about being early or late, he minds being a little bit early or late much less than he minds being a lot early or late: his loss is the square of the number of minutes before or after noon that he arrives. How many minutes before noon should Joe’s friend leave to minimize his expected loss from being early or late? (Please round your answer to the nearest whole minute, like 12.)
Flag question: Question 5Question 51 pts
You operate an ice cream stand at a summer festival. Each day, you must decide how many ice cream bars to stock, knowing demand is uncertain. The daily demand is normally distributed with a mean of 200 bars and a standard deviation of 50 bars.
- Each bar costs you $1 to stock.
- If demand exceeds your stock, you lose $3 per unsatisfied customer (missed sale + reputational cost).
- If you overstock, leftover bars melt and must be discarded, costing you $1 per unsold bar.
Your goal is to minimize expected cost (from both lost sales and melted surplus).
How many ice cream bars should you stock each day to minimize expected cost?
(Please enter your answer as a whole number of bars.)
Flag question: Question 6Question 61 pts
In Problem 5, For what standard deviation of the daily demand would it be optimal to stock 220 bars? (Please round your answer to 2 digits, like 12.34.)
Flag question: Question 7Question 71 pts
In Problem 5 suppose that the distribution of daily demand is Poisson with mean 200 bars/day. Now what is the optimal number of bars to stock? (Please enter your answer as a whole number of bars.)
Flag question: Question 8Question 81 pts
An airline operates a plane with 150 seats. The airline sells seats on the plane (reservations) for $500. The probability that a passenger with a reservation will show up is uncertain. It has a prior distribution that is uniformly distributed between 0.8 and 1, denoted p ~ U[0.8, 1]. If the airline overbooks (sells more reservations than there are seats) and more than 150 passengers show up, then the excess number have to catch a later flight, and each such excess (“bumped”) passenger costs the airline $1500. Passengers who don’t show up lose their $500 (the airline keeps it).
What is the smallest value of the show-up probability p for which it is optimal for the airline not to overbook? (Please round your answer to 3 digits, like 0.123. Assume that the airline maximizes EMV.)
Flag question: Question 9Question 91 pts
In Problem 8, if the show-up probability is p = 0.99, then what is the smallest cost to the airline for each excess(“bumped”) passenger that would make it optimal for the airline not to overbook? (Please round your answer to 3 digits, like 0.123. Assume that the airline maximizes EMV.)
Flag question: Question 10Question 101 pts
In Problem 8, if the show-up probability is p = 0.9 and each excess (“bumped”) passenger costs the airline $1500, then what is the optimal number of reservations for the airline to sell? (Please enter your answer as an integer, like 12.)