MATH 1081 UO Mathematical Methods
for Data Analytics 2
Continuous Assessment 1.2: Problem Solving Exercise
Instructions:
• Structure of the assessment: This assessment is worth 25% of your final
grade and is due no later than 12 pm on Monday, Week 9. This assessment
consists of 9 questions which ca
y 100 marks.
• Save your work: The file you submit needs to be in a pdf format, and save
your submission as “your student ID Assessment 1.2 MATH1081.pdf”.
• Use of R: You can use R if you prefer. If you use R, you must provide you
R codes to get full marks. Upload your R script and screenshot the R codes in
your answer sheet.
• Show your work: Show all necessary steps so that the reader can follow you
solution procedure.
• Presentation: Write your solutions clearly in a well-organised manner. Use
notations accurately. Label the questions and sub-questions. You may not need
to copy the questions again to your answer sheet. However, you can do it if you
wish.
• Writing your solutions: You have 3 options to write your answers.
1. digital writing: If you have a touchscreen or tablet device, then you can
use Microsoft Inking to record your answers. Click Here for a video uses a
Wacom tablet to show you this technique.
2. equation editor: If you prefer to type your equations manually using an
equation editor then you can use the inbuilt Microsoft Word editor. The
Microsoft Help Resources contain more information if you get stuck.
1
https:
unisa.au.panopto.com/Panopto/Pages/Embed.aspx?id=2b5916e0-d0c8-4c9b-87f4-ac24003f0db6&
3. handwriting: If you prefer, you can use a pen/pencil and paper to hand-
write your equations and then digitize them using a scanner or by taking a
photo on your phone. Take care to make sure the final image is readable.
Do not forget to set up your pictures so that the marker does not
need to spend time a
anging your photos (you will penalize a few
marks for presentation if the teaching team has to do it for you)
• Acknowledgement of work: When submitting online, you acknowledge that
the submitted assignment is your own work unless otherwise stated.
• Academic integrity: The University’s policy on academic misconduct will be
strictly applied. Here are some tips to avoid academic misconduct:
– Do not copy from any printed or electronic source or from any person.
– Write your own solutions. You may discuss your work with others, but
you must write up your solutions yourself. You are not allowed to use some-
one else’s written work when writing up your submission.
– Do not give inappropriate help. Giving inappropriate help is just as
serious as receiving it and will have the same consequences. Do not show
your completed exercise to others. Dispose of drafts so that no one can access
them.
– Acknowledge help and joint work. If you receive any help from anothe
source (for example, students, tutors, friends, internet), you must make a
note of it on your submission.
• Late submission: Any late submission will attract a penalty of 5 marks avail-
able per day for five days. The cut-off time is 12 pm each day. After five
days from the assessment due date, no submissions will be marked, and zero
marks will be granted.
2
1. Let
dy1
dt
= y′1 = 3y2
dy2
dt
= y′2 = −y1 + 4y2,
where y1(0) = 1 and y2(0) = 1. Find the solution of the system and comment on
its stability. [10 marks]
2. Three sky-jumpers attempt independently of each to land on a straight line on the
ground marked ABC, where B is the half-way point of the line AC. Each jumpe
is able to land somewhere on the line with probability 0.8. Otherwise the jumpe
ends up landing some where away from the line. If any one of the jumper happens
to land on the line, then he/she will land in the AB section with probability 0.5,
or in the BC section also with probability 0.5.
(a) What is the probability that any 2 out of the 3 jumpers manage to land
somewhere on the line? [2 marks]
(b) What is the probability that any 2 out of the 3 jumpers manage to land
somewhere on the AB section of the line? The other one could have eithe
landed in the BC section or not on the line at all. [2 marks]
(c) Jenny is one of the jumpers. What is the probability that Jenny is one of
only two jumpers who manage to land somewhere on the line? [3 marks]
(d) What is the probability that two jumpers manage to land somewhere on the
line, given that Jenny is one of them? [3 marks]
[10 marks]
3. For a particular COVID test, there is a probability of 0.2 that the test result
comes out positive if it is randomly applied to a randomly chosen person off the
street in Gotham City. Also in the same city, the probability that the person
actually has COVID if tested positive is 0.90 and the probability that the person
actually does not have COVID if tested negative is 0.99. Assume that everyone
has the same probabilities of having COVID and testing positive for COVID.
(a) If I randomly pick a person in Gotham City, what is the probability that the
person actually does not have COVID? [6 marks]
(b) What is the probability that the test result is positive given that the person
actually has COVID? [3 marks]
(c) What is the probability that the test result is negative given that the person
actually does not have COVID? [3 marks]
[12 marks]
3
4. A fair coin is tossed 3 times. Let X = (X1, X2) where X1 counts the number of
heads in the 3 tosses, and X2 counts the number of tails in the 3 tosses.
(a) Write down the sample space S. [1 mark]
(b) Show the mapping of each sample point in S to the plane R2 by the random
variable X. [2 marks]
(c) Tabulate the joint probability mass function of X1 and X2. [2 marks]
(d) Calculate the marginal probability mass function of X1 and X2. What can
you deduce about them? Will each co-ordinate have the same values fo
their mean and variance? [2 marks]
(e) Calculate ρX1,X2 , the co
elation coefficient. Are they independent?
- [5 marks]
[12 marks]
5. Suppose X1 ∼ Bin(m, p) and X2 ∼ Bin(n, p) and are independent. E.g., X1
counts the number of heads for the 1st m independent coin tosses and X2 counts
the number of heads for the next n independent coin tosses. The probability of
getting heads for any toss is p where 0 ≤ p ≤ 1. Let S = X1 +X2.
(a) Using
pS(s) = P [S = s] =
s∑
x1=0
P [X1 = x1]P [X2 = s− x1],
obtain the probability mass function of S. Here Cba ≡ 0 if a > b.
Note: You will need the identity
∑k
y=0C
m
y C
n
k−y = C
m+n
k for y a non-negative
integer. [3 marks]
(b) What do you observe? [1 mark]
(c) Obtain an expression fo
P [X1 = x1|S = s],
where x1 ≤ s and x1 is a non-negative integer. [4 marks]
(d) Obtain an expression fo
P [S = s|X1 = x1],
where x1 ≤ s and x1 is a non-negative integer. [4 marks]
[12 marks]
4
6. Suppose X has the probability density function
fX(x) =
k(x+ 1)2, for − 1 ≤ x ≤ 0,
k(1− x)2, for 0 ≤ x ≤ 1,
0 otherwise.
(a) Find k so that fX(x) is a probability density function. [3 marks]
(b) Derive the cumulative distribution function FX(x). [3 marks]
(c) Calculate P [−0.5 ≤ X ≤ 0.5] by integration. [3 marks]
(d) Calculate P [−0.5 ≤ X ≤ 0.5] by using the cumulative distribution function.
- [1 mark]
(e) Calculate E[X] and Var[X]. [6 marks]
[16 marks]
7. The number of visitors clicking an online shopping website is distributed as Pois-
son with a mean visit rate of 7 per hour. Any individual visitor who goes onto the
website has a 65% chance of making a purchase. Assume each individual visito
acts independently of each other, and assume that the visitor only makes 1 click
to the website within that hour.
(a) What is the probability that within a particular hour, there are 5 visits to
the site? [2 marks]
(b) What is the probability that given n visits to the site, there are x purchases
where x = 0, 1, 2, · · · , n? [2 marks]
(c) What is the probability that there are 5 visits to the site with only 3
purchases? [2 marks]
(d) Simulate using R the number of actual purchases over 10 hour period.
- [5 marks]
[11 marks]
8. Refer to the previous question. The number of visitors clicking an online shopping
website is distributed as Poisson with a mean visit rate of 7 per hour. Any indi-
vidual visitor who goes onto the website has a 65% chance of making a purchase.
Assume each individual visitor acts independently of each other, and assume that
the visitor only makes 1 click to the website within that hour. If the ith visito
makes a purchase, the log of the amount spent ln(Yi) is normally distributed as
N(3, 2). Simulate the number of visitors in a one hour period, the number from
these visitors who make a purchase, and if they make a purchase, their purchase
amounts. [5 marks]
5
9. Look at the treering data in R . This give the standardised values of annual
treering outward growth of a particular tree from 6000 BC to 1979 AD. Calculate
a 99% confidence interval of the mean standardised value of the annual treering
growth. [6 marks]
10. Look at the sleep data in R . The extra column gives the hours of extra sleep of
that particular participant in this study. Test the hypothesis at the 1% significance
level that the extra hours of sleep is 0 hours. [6 marks]
6