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STAT2003 Mathematical Probability
Exam type Online, non-invigilated, final examination
Exam technology File upload to Blackboard Assignment
Exam date and time
Your examination will begin at the time specified in your personal examination timetable.
If you commence your examination after this time, the end for your examination does
NOT change.
The total time for your examination from the scheduled starting time will be:
2 hours 10 minutes (including 10 minutes reading time during which you should read the
exam paper and plan your responses to the questions).
A 15-minute submission period is available for submitting your examination after the
allowed time shown above. If your examination is submitted after this period, late
penalties will be applied unless you can demonstrate that there were problems with the
system and/or process that were beyond your control.
Exam window
You must commence your exam at the time listed in your personalised timetable. You
have from the start date/time to the end date/time listed in which you must complete your
exam.
Permitted materials
This is a closed book exam – only specified materials are permitted. You may use all
material from the STAT2003/STAT7003 blackboard site, including the Lecture Notes.
Python or Mathematica may be used for calculations. You may not make use of any
other material, including web sites or books.
Recommended
materials
Ensure the following materials are available during the exam:
Calculator/computer, phone/camera/scanner
Instructions
You will need to download the question paper included within the Blackboard Test. Once
you have completed the exam, upload the completed exam answers file to the
Blackboard assignment submission link. You may submit multiple times, but only the last
uploaded file will be graded.
Write your answers on blank paper, or write electronically on a suitable device. Scan or
photograph your work if necessary and upload your answers to Blackboard as a single
PDF file, before the end of the allowed time.
Include your name and student number on the first page of the file that
you submit, and clearly label your solutions so that it is clear which problem it is a
solution to.
Who to contact
Given the nature of this examination, responding to student queries and/or relaying
corrections to exam content during the exam may not be feasible.
If you have any concerns or queries about a particular question or need to make any
assumptions to answer the question, state these at the start of your solution to that
Semester One Final Examinations, 2021 STAT2003 Mathematical Probability
Page 2 of 5
question. You may also include queries you may have made with respect to a particular
question, should you have been able to ‘raise your hand’ in an examination-type setting.
If you experience any interruptions to your examination, please collect evidence of the
interruption (e.g. photographs, screenshots or emails).
If you experience any issues during the examination, contact ONLY the Library AskUs
service for advice as soon as practicable:
Chat: support.my.uq.edu.au/app/chat/chat_launch_lib
Phone: +61 7 3506 2615
Email: [email protected]
You should also ask for an email documenting the advice provided so you can provide
this as evidence for a late submission.
Late or incomplete
submissions
In the event of a late submission, you will be required to submit evidence that you
completed the assessment in the time allowed. This will also apply if there is an error in
your submission (e.g. corrupt file, missing pages, poor quality scan). We strongly
recommend you use a phone camera to take time-stamped photos (or a video) of every
page of your paper during the time allowed (even if you submit on time).
If you submit your paper after the due time, then you should send details to SMP Exams
([email protected]) as soon as possible after the end of the time allowed. Include
an explanation of why you submitted late (with any evidence of technical issues) AND
time-stamped images of every page of your paper (eg screen shot from your phone
showing both the image and the time at which it was taken).
Important exam
condition
information
Academic integrity is a core value of the UQ community and as such the highest
standards of academic integrity apply to all examinations, whether undertaken in-person
or online.
This means:
• You are permitted to refer to the allowed resources for this exam, but you cannot
cut-and-paste material other than your own work as answers.
• You are not permitted to consult any other person – whether directly, online, or
through any other means – about any aspect of this examination during the
period that it is available.
• If it is found that you have given or sought outside assistance with this
examination, then that will be deemed to be cheating.
If you submit your online exam after the end of your specified reading time, duration, and
15 minutes submission time, the following penalties will be applied to your final
examination score for late submission:
• Less than 5 minutes – 5% penalty
• From 5 minutes to less than 15 minutes – 20% penalty
• More than 15 minutes – 100% penalty
These penalties will be applied to all online exams unless there is sufficient evidence
of problems with the system and/or process that were beyond your control.
Semester One Final Examinations, 2021 STAT2003 Mathematical Probability
Page 3 of 5
Undertaking this online exam deems your commitment to UQ’s academic integrity
pledge as summarised in the following declaration:
“I certify that I have completed this examination in an honest, fair and trustworthy
manner, that my submitted answers are entirely my own work, and that I have neither
given nor received any unauthorised assistance on this examination”.
Semester One Final Examinations, 2021 STAT2003 Mathematical Probability
1. [Total: 11 marks] The Yummy Snacks® company sells bags of coloured peas. Each bag contains
anything from 0 to 10 peas, according to a discrete uniform distribution on {0, 1, . . . , 10}. Each pea
comes in one of three equally likely colours: red, green, or blue. Let N be the total number of peas
in a bag, and let R,G,B be the number of red, green, and blue peas in that bag; in particular, N =
R +G+B.
(a) [2 marks] Give the conditional probability mass function of R given N = 5.
(b) [3 marks] Compute the expected number of red peas in a bag.
(c) [3 marks] Suppose we buy 3 bags. What is the probability that all bags are empty?
(d) [3 marks] What is P(R = 2, G = 2, B = 2)?
2. [Total: 11 marks] In April 2021, the European Medicines Agency received reports of 169 cases of a rare
brain blood clot in people who received a certain COVID-19 vaccine, out of 34 million administered
doses of the vaccine. Suppose 100,000 people in Australia will receive the vaccine in the next month.
Let N be the number of people, out of these 100,000, who will develop a brain blood clot after the
vaccination.
(a) [3 marks] Under what assumptions would a Binomial distribution for N be appropriate?
(b) [3 marks] Under this Binomial model, what is the probability that none of these 100,000 vacci-
nated people will develop a blood clot?
(c) [5 marks] We can also use a Poisson model for the distribution of N . Give the corresponding rate
parameter λ and compute the probability P(N ≤ 2) under this model.
3. [Total: 12 marks] Let X1, X2, . . . be an iid sequence of random variables with probability density
function f , given by
f(x) =
{
c(θ − x)x, if 0 ≤ x ≤ θ,
0 otherwise,
where θ > 0. Let Sn = X1 + · · ·+Xn for n = 2, 3, . . ..
(a) [3 marks] Express the constant c in terms of the parameter θ.
(b) [4 marks] For θ = 2 compute EX1 and Var(X1).
(c) [5 marks] Using the central limit theorem, approximate P(S10 > 6θ) in terms of the cdf Φ of the
standard normal distribution.
4. [Total: 11 marks] Consider the following acceptance–rejection algorithm to simulate a random variable
Z.
1. Generate U ∼ U(0, 1).
2. Set X = −2 lnU .
3. Generate Y ∼ U(0, e−X/2).
4. If Y ≤ e−X , return Z = X; otherwise, return to Step 1.
(a) [4 marks] Specify the probability distribution (e.g., give the pdf or cdf) of Z.
(b) [3 marks] What is the efficiency (i.e., acceptance probability) of this algorithm?
(c) [4 marks] Formulate an inverse-transform algorithm to generate Z.
Page 4 of 5
Semester One Final Examinations, 2021 STAT2003 Mathematical Probability
5. [Total: 12 marks] Let B0, B1, . . . be a sequence of independent Ber(p) random variables. Consider a
Markov chain {Xn, n = 0, 1, 2, . . .} on the state space {0, 1, . . . , 5} defined by X0 = 0 and
Xn+1 =
Xn + 2Bn − 1 if 0 ≤ Xn + 2Bn − 1 ≤ 5,
0 if Xn + 2Bn − 1 < 0,
5 if Xn + 2Bn − 1 > 5.
(a) [3 marks] Give the probability mass function of the random variable 2Bn − 1.
(b) [3 marks] Draw the transition graph of this Markov process.
(c) [3 marks] Specify the one-step transition matrix P for this Markov chain.
(d) [3 marks] Sketch (by hand) a typical realization of Xn, n = 1, . . . , 10 for the cases p = 1/2 and
p = 0.1.
6. [Total: 11 marks] The lifetime T of some electrical component has a failure rate h that increases
linearly with time:
h(t) = α t, t ≥ 0,
where α > 0.
(a) [4 marks] Find the cumulative distribution function (cdf) of the lifetime T of the component.
(b) [3 marks] Conditional on T > 1, what is the probability that T > 2?
(c) [4 marks] Suppose that n such components are placed in a series system and that the lifetimes are
independent of each other. Compute the expected lifetime of the series system.
