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The University of Nottingham 
Suggested time to complete: TWO Hours THIRTY Minutes 
Paper set: 19/05/2020 - 10:00 
Paper due: 26/05/2020 - 10:00 
Answer ALL questions 
Your solutions should be written on white paper using dark ink (not pencil), on a tablet, or 
typeset. Do not write close to the margins. Your solutions should include complete 
explanations and all intermediate derivations. Your solutions should be based on the material 
covered in the module and its prerequisites only. Any notation used should be consistent with 
that in the Lecture Notes. 
Guidance on the Alternative Assessment Arrangements can be found on the Faculty of Science 
Moodle page: https://moodle.nottingham.ac.uk/course/view.php?id=99154#section-2 
Submit your answers as a single PDF with each page in the correct orientation, to the 
appropriate dropbox on the module’s Moodle page. Use the standard naming 
convention for your document: [StudentID]_[ModuleCode].pdf. Please check the 
box indicated on Moodle to confirm that you have read and understood the statement 
on academic integrity: https://moodle.nottingham.ac.uk/pluginfile.php/6288943/mod_ 
A scan of handwritten notes is completely acceptable. Make sure your PDF is easily readable 
and does not require magnification. Text which is not in focus or is not legible for any other 
reason will be ignored. If your scan is larger than 20Mb, please see if it can easily be reduced 
in size (e.g. scan in black white, use a lower dpi — but not so low that readability is 
Staff are not permitted to answer assessment or teaching queries during the assessment 
period. If you spot what you think may be an error on the exam paper, note this in your 
submission but answer the question as written. Where necessary, minor clarifications or 
general guidance may be posted on Moodle for all students to access. 
Students with approved accommodations are permitted an extension of 3 days. 
The standard University of Nottingham penalty of 5% deduction per working day will 
apply to any late submission. 
MATH4007-E1 Turn over 
Academic Integrity in Alternative Assessments 
The alternative assessment tasks for summer 2020 are to replace exams that would have 
assessed your individual performance. You will work remotely on your alternative assessment 
tasks and they will all be undertaken in “open book” conditions. Work submitted for 
assessment should be entirely your own work. You must not collude with others or employ the 
services of others to work on your assessment. As with all assessments, you also need to avoid 
plagiarism. Plagiarism, collusion and false authorship are all examples of academic misconduct. 
They are defined in the University Academic Misconduct Policy at: https://www.nottingham.ac. 
Plagiarism: representing another person’s work or ideas as your own. You could do this by 
failing to correctly acknowledge others’ ideas and work as sources of information in an 
assignment or neglecting to use quotation marks. This also applies to the use of graphical 
material, calculations etc. in that plagiarism is not limited to text-based sources. There is 
further guidance about avoiding plagiarism on the University of Nottingham website. 
False Authorship: where you are not the author of the work you submit. This may include 
submitting the work of another student or submitting work that has been produced (in whole 
or in part) by a third party such as through an essay mill website. As it is the authorship of an 
assignment that is contested, there is no requirement to prove that the assignment has been 
purchased for this to be classed as false authorship. 
Collusion: cooperation in order to gain an unpermitted advantage. This may occur where you 
have consciously collaborated on a piece of work, in part or whole, and passed it off as your 
own individual effort or where you authorise another student to use your work, in part or 
whole, and to submit it as their own. Note that working with one or more other students to 
plan your assignment would be classed as collusion, even if you go on to complete your 
assignment independently after this preparatory work. Allowing someone else to copy your 
work and submit it as their own is also a form of collusion. 
Statement of Academic Integrity 
By submitting a piece of work for assessment you are agreeing to the following statements: 
1. I confirm that I have read and understood the definitions of plagiarism, false authorship 
and collusion. 
2. I confirm that this assessment is my own work and is not copied from any other person’s 
work (published or unpublished). 
3. I confirm that I have not worked with others to complete this work. 
4. I understand that plagiarism, false authorship, and collusion are academic offences and I 
may be referred to the Academic Misconduct Committee if plagiarism, false authorship or 
collusion is suspected. 
MATH4007-E1 Turn over 
1 MATH4007-E1 
1. (a) i) The truncated Poisson dsitribution has probability mass function 
(; ) = 
!(1 − −) 
, = 1, 2, 3,… , 
where > 0 is a parameter. Prior information about is summarized by 
() ∝ −|−3|, > 0. 
Given observed data 1 = 2, 2 = 5, 3 = 13, derive the log posterior distribution, 
denoted by (), up to an additive constant. 
ii) It is required to find the maximum of (). An initial interval thought to contain a 
maximum is given by 1 = 4.36, 3 = 5.81. Carry out two iterations of the Golden 
Ratio method, i.e. find the next two intervals containing a maximum of (). 
iii) What is the statistical interpretation of the output of the algorithm? 
[15 marks] 
(b) The joint density of two random variables and is given by 
(, ) ∝ 22 exp(− − 5 − 4), , > 0. 
i) Derive the Laplace approximation to the marginal density (). 
ii) Evaluate this for the case = 2. 
iii) Treating as missing information, give full details of how the EM algorithm can be 
used to find the mode of the marginal distribution (). 
iv) Starting from an initial value (0) = 0.1, perform two iterations of the EM algorithm. 
[25 marks] 
2 MATH4007-E1 
2. (a) It is required to sample from a density , where 
() = 
(1 − ) 
, 0 < < 1, 
and > 0 is a constant. 
i) Find the constant . 
ii) Hence, explain how to sample from using inversion. 
iii) Produce one sample from , given a sample = 0.4 from a (0, 1) distribution. 
[8 marks] 
(b) The joint density of two random variables and is given by 
(, ) ∝ 22 exp(− − 5 − 4), , > 0. 
i) Show that the marginal density of is proportional to 
( + 4)3 
ii) The density of a random variablewhich follows a Gamma distribution with parameters 
and is 
() ∝ −1 exp{−}. 
Show how samples from the marginal distribution of can be obtained using the 
rejection algorithm, using samples from a Gamma distribution with = 3, = 5. 
iii) Assume that samples from the conditional distribution |(|) can be obtained 
for any value of . (You do not have to find this distribution or how to sample from 
it.) Explain how this and the above result can be used to sample from the joint 
density of and to estimate [ ]. 
[20 marks] 
(c) i) Consider the Nearest Neighbour estimator of a density . Carefully describe 
the rationale behind this estimator, and discuss the influence of on the resulting 
ii) Three data points from an unknown density are observed, 1 = 1, 2 = 2 and 
3 = 4. Calculate the Nearest Neighbour estimator of , with = 2. 
[12 marks] 
MATH4007-E1 Turn Over 
3 MATH4007-E1 
3. (a) Consider the density 
(, ) ∝ 22 exp{− − 5 − 4}, > 0, > 0. 
i) Find the full conditional distributions (|) and (|). 
ii) Hence, describe a Gibbs sampler to sample from . 
iii) Suppose instead that the Metropolis-Hastings algorithm is to be used to sample 
from . Describe fully a random walk Metropolis algorithm which updates both 
variables simultaneously, using proposals of the form 
As part of your answer, discuss the role of on the performance of the sampler. 
iv) Suppose = 2, the identity matrix, and the chain is currently in the state = 2, 
= 1. Given 3 independent (0, 1) random numbers 
= 0.2, = 0.8, = 0.45, 
perform one update of the chain described in the previous part. 
[28 marks] 
(b) The following data are available, which are believed to be random samples from a 
population with mean = 7 
9.1 5.8 5.1 9.7 5.5 4.3 6.0 
i) Explain why a randomisation test might be preferred to a t-test in order to test the 
hypothesis 0 ∶ = 7. 
ii) Describe a suitable randomisation test to test0 ∶ = 7, stating any assumptions 
you make. 
iii) Use the (0, 1) random numbers 
{0.46, 0.84, 0.02, 0.76, 0.67, 0.53, 0.22} 
in order to calculate one replicate of your test statistic for the test in (ii). 
[12 marks] 
MATH4007-E1 END 
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