QUESTION 1 [34 marks] Maximizing Pawfits
Pavlov found the ideal manufacturing plant with your help from the previous assignment. Pavlov recruited
his younger brother, Markov
, to help him with the manufacturing
process. However, Markov needs your
help identifying the optimal manufacturing process to maximize profits from the Pavlovian conditioner sales.
Each unit of Pavlovian conditioner has its own measure of quality
, which has the probability density
a) [8 marks] The constant
depends on the manufacturing process, and Markov can alter it.
(i) [2 marks] Which values of
is a valid probability density function?
(ii) [2 marks] Determine
, the expected value of the quality of a unit of conditioner.
(iii) [4 marks] Set the seed to your student ID. Using
, simulate the quality of
the Pavlovian conditioner when
Print the quality of the first 6 units of conditioner and the mean
quality of all
[8 marks] Markov inspects every unit and any unit for which
is discarded, while all other products
would be sold.
(i) [4 marks] Find the probability that a randomly chosen Pavlovian conditioner product is discarded.
(ii) [4 marks] By using your simulated data from part a) (iii), calculate the proportion of units that get
discarded in the simulation.
c) [18 marks] Pavlov’s costs (in dollars) of manufacturing each unit of product are given by