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MAS6012 Semester 2 Project
This project contributes 15% towards your module mark for MAS6012. Submit a single pdf on the MAS370
MOLE page. The deadline for submission is Tuesday 3rd March 2020 at noon. You must include all
your working and present your solutions clearly. Use R Markdown for this assignment including all your R
code in the pdf (use echo = T in the code chunk).
Your submitted solutions must be entirely your own work; do not work with anyone else on this project.
Your mark will be out of 100. Some answers to questions in this project can be computed directly, some will
require Monte Carlo simulation. Use exact methods where possible.
1. A computer model is given by the function
Y = X2
1 + 2X2X3.
The true values of the three inputs are uncertain, with X1 ≥ N(6, 9), X2 ≥ U[9, 14] and X3 ≥ N(9, 1),
with X1, X2, X3 independent. The model user can choose to learn the true value of one of the inputs.
The model user wishes to reduce the variance of the output Y as much as possible. Calculate the
expected reduction in variance by learning each of X1, X2 and X3 separately and hence suggest which
input the user should choose to learn and why.
2. By considering the input distributions of X1, X2 and X3 and the computer model itself, explain the
relative sizes of the three expected reductions in variance observed in Part 1.
3. Suppose instead that X1 now has a gamma, Ga(–,—), distribution with shape and rate parameters
– = 12 and — = 2 respectively. Note that the probability density function for a random variable X
distributed according to a gamma distribution with shape parmeter – and rate parameter — is
fX(x) = —–
(–)
x–≠1 exp (≠—x).
X2 and X3 keep the same distribution as in Part 1. By calculating the expected reduction in variance,
suggest which input the user should now choose to learn and why.
4. Explain any dierences
in your answers to Parts 1 and 3. Does replacing the N(6,9) distribution with
the Ga(–=12,—=2) distribution for X1 raise any concerns?
5. Now suppose that there is a single input, X, and the output, Y , is given by
Y = 2X + 3
where X is uncertain with X ≥ N(–, 9) and – is a random variable distributed according to – ≥ N(6, 2).
What is the reduction in the variance of Y from knowing the value of –? Why do we not need to
calculate the expected reduction in the variance of Y?
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