Copyright Note to students: Copyright of this assessment brief is with UCL and the module leader(s) named on page 2.  If this brief draws upon work by third parties (e.g. Case Study publishers) such third parties also hold copyright. It must not be copied, reproduced, transferred, distributed, leased, licensed or shared with any other individual(s) and/or organisations, including web-based organisations, without permission of the copyright holder(s).

Academic Misconduct: Academic Misconduct is defined as any action or attempted action that may result  in  a  student  obtaining  an  unfair  academic  advantage. Academic  misconduct  includes plagiarism,   obtaining   help   from/sharing   work   with   others   be   they   individuals   and/or organisations or any other form of cheating. Penalties for Academic Misconduct vary, up to and including automatic expulsion from UCL for those found to have engaged in contract cheating (students commissioning  others to  complete work  on their behalf. Definitions  and penalties  for Academic Misconduct are outlined in the Academic Manual.

Referencing: In an exam no external research would normally take place and therefore the matter of references does not arise. This applies to this online assessment. Whilst you are encouraged to draw upon  your  learning  from  module  materials,  including  lecture  slides, books,  articles,  you  should communicate your learning in your own words and, where appropriate, calculations. However, if you were to make direct use of somebody else’s work, either by memorising it or copying it word for word, you must reference and provide full citation for ALL sources used, including articles, text books, lecture slides and module materials. This includes any direct quotes and paraphrased text. If in doubt, reference it. If you need further guidance on referencing please see UCL’s referencing tutorial for students here:

https://library-guides.ucl.ac.uk/referencing-plagiarism/welcome.  Failure  to  cite  references  correctly may result in your work being referred to the Academic Misconduct Panel.

Examination length: THREE (3) hours

There is ONE (1) section to the examination paper. This section consists ofNINE (9) questions worth ONE HUNDRED (100) marks. Candidates should attempt ALL questions.

There is no formula sheet provided.

You are advised to allocate your time between different questions in proportion to the marks available.

Module Leader: Rouba Ibrahim and Deyu Ming

Internal Assessor: Andrew Whiter

1.  [25 points] Social distancing is an important measure to prevent the spread of COVID- 19. Assume that the chance, denoted by r, that a healthy person gets infected by a virus carrier who is L metres away is modelled by r = exp{−L2 /γ2 }, where γ > 0.

(a)  [2 points] Discuss mathematically how r changes as L and γ change in the specified model.

(b)  [4 points] Give your practical interpretation of γ and list at least FOUR real-world factors that you think may affect the magnitude of γ .

(c)  [7 points] Assume that a virus carrier is at location c on a road with infinite length, and a healthy person is on the same road at location X, which is normally dis- tributed with mean µ and standard deviation σ . Calculate the expected chance that the healthy person will get infected.

(d)  [8 points] Following the Government’s two metre social distancing guidance, we assume that the distance between the location of the virus carrier and the expected location of the healthy person is 2 metres. Find the value of γ such that the maxi- mum expected chance that the healthy person gets infected is 10%.

(e)  [4 points] Discuss and justify at least two limitations of the specified model for r.

2.  [10 points] We define a Markov chain {Yn , n ≥ 0} in the following manner.  Yn  can take values in the state space {−3, −2, −1, 0, 1, 2, 3}. We assume that Y0  = 0. We also assume the following transition probabilities:

P(Yn+1  = i + 1|Yn  = i) = P(Yn+1  = i − 1|Yn  = i) = 1/2 for i ∈ {−2, −1, 0, 1, 2, },

and

P(Yn+1  = 2|Yn  = 3) = P(Yn+1  = −2|Yn  = −3) = 1.

(a)  [5 points] Is the corresponding sequence of absolute values, {|Y0 |, |Y1 |, ...}, also a Markov chain? Justify your answer and, if your answer is affirmative, write out the transition probabilities of the chain.

(b)  [5 points] Consider the function f defined as follows:

!1

"

f(x) = # −1

"

if x > 0

if x < 0

if x = 0.

Is {f(Y0 ), f(Y1 ), ...} a Markov chain? Justify your answer and, if your answer is affirmative, write out the transition probabilities of the chain.

3.  [7 points] Consider two independent random variables, S and T, which are both expo- nentially distributed with the same rate λ . Determine the probability density functions of the following random variable: R = S + T.

4.  [10 points] Rouba and Deyu are meeting for a coffee at a local Bloomsbury cafe. Rouba arrives at a time which is uniformly distributed between 9 AM and 10 AM, and Deyu independently arrives at a time which is uniformly distributed between 9AM and 11AM.