MATH268: Assignment 1
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MATH268: Assignment 1
Exercise 1 A TV set factory produces 8 (thousands of) units of a certain product per day with a proportion of defect TV sets Q . We know that Q has a probability distribution function given by
f (q) = 
A defect TV set leads to a loss of £50 . A good TV set produces a profit of £5 . By using the expected value criterion, how many TV sets should the manufacturer produce per day to minimise the loss (maximise the profit)? [20 marks]
Exercise 2 Suppose that two producers M and N provide lots of five fridge shelves necessary for Indesit to add to their fridges before sale. Suppose that 75% of the shelves come from vendor M and that 1% of the shelves from vendor M do not fit well. The shelves from vendor N include 2% which do not fit well. Suppose a lot is picked up and checked whereas the shelves fit. After inspection, only one shelf is found not to fit.
(a) Determine the posterior probability that the sampled lot of fridge shelves was delivered
from vendor M. [15 marks]
(b) Determine the posterior probability that the sampled lot of fridge shelves was delivered
from vendor N. [5 marks]
Exercise 3 An airplane rental company is attempting to determine whether to buy various second hand airplanes to use them for business. The rental company is concerned that the airplanes might not be reliable. Assume that the decision will be made on the basis of obtaining the highest financial profit or minimising the loss. Each airplane which is bought and is defective costs the company ↔100,000, and each airplane which is not defective will contribute ↔10,000 to the company profit. Assume that 10% of all airplanes are defective. The company may buy the airplane, refuse to buy the airplane, or test the airplane before deciding to buy.
If costs £100 to test an aiplane; the test result is either reliable or non reliable. If the test result indicates non reliable, the airplane will definitely break. However, 20% of the airplanes which are defective are deemed reliable. A reliable aiplanee will always test reliable.
The companyt’s goal is to maximize the expected economic benefit (net profit) . Using a diagram tree, determine the optimal policy of the company. [30 marks]
Exercise 4 Mary is an athlete with a half maratone scheduled for next week. She has just received an invitation to a wedding which is held 1 days before the competition. Her performance at the competition depends on how she spends the day before the competition.
Mary has 3 options.
(a3) spending every day before the competion training as hard possible. This implies refusing to go to wedding.
(a2) train hard everyday including the morning of the wedding day but attend the wedding.
(a1) attend the wedding and skip any training on the day of the wedding.
She does not know who is going to participate at the competition but there are 3 possibilities. The level of compeition can be easy if only local competitors participate, it can be of medium difficulty if national competitors come, or hard if many international participants attend it.
On a scale of 0 to 100, Mary anticipates her performance at the competition to depend of her decision to train in the next days and the participation at the competition as
Easy Medium Hard
|
65 |
40 |
20 |
|
72 |
65 |
61 |
|
80 |
68 |
62 |
(a) Determine the optimal decision of Mary, in case (i) she applied the most pessimistic
criterion, (ii) she uses the Laplace criterion, (iii) she wants to minimise the regret. [3+3+4=10 marks]
(b) Suppose Mary only cares about the place at the competition. She gets a High classifiction
if the perfomance is above (and including) 70; a Medium-High classification if the perfor- mance is above (and including) 60; a Medium classification if performace is above (and including) 50; a Medium-Low classification if the performance is above (and including) 40 and a Low classification if the performance is strictly below 40 . Mary prefers a High classification.
Determine the optimal decision of Mary, in case (i) she applies the most optimistic cri- terion, (ii) she wants to minimise regret. [10+10=20 marks]
2022-10-19