MATH2801: Theory of Statistics Assignment
Submission date: Please follow the instructions below for completing the assignment
(worth 10% of the final mark):
1. This assignment is a group assignment (maximum of 4 people). You can of course
work on the assignment by yourself if you wish.
2. Each person of the group must submit an electronic copy of the assignment solution
via Turnitin on Moodle before 11am on Friday 2nd August (Week 9).
A single paper copy of the assignment (per group) is to be handed to lecturer
(Dr Jakub Stoklosa) before the end of the Friday’s lecture 11am on
Friday 2nd August (Week 9).
3. The assignment must be typed and submitted as a pdf file. Please make sure all
the names and zIDs of each group member are written on the first page of the
Length: No more than 6 pages, including this cover sheet.
I (We) declare that this assessment item is my (our) own work, except where acknowledged,
and has not been submitted for academic credit elsewhere, and acknowledge that the assessor
of this item may, for the purpose of assessing this item:
Reproduce this assessment item and provide a copy to another member of the University;
Communicate a copy of this assessment item to a plagiarism checking service (which
may then retain a copy of the assessment item on its database for the purpose of future
I (We) certify that I (We) have read and understood the University Rules in respect of
Student Academic Misconduct.
1. This question requires the use of RStudio but you do not need to include the RStudio
commands used for this question.
For this question you are required to construct and print out six graphs and to include
them with your written answers to the questions. These six graphs should be produced
and printed together on one A4 page. You can do this using a function in RStudio
called par(mfrow = c(3, 2)) (this command produces a 3 by 2 grid of plots).
The graphs should be clearly titled and the axes labelled.
The ocean swell produces spectacular eruptions of water through a hole in the cliff at
Kiama, about 120km south of Sydney, known as the Blowhole. The times at which 65
successive eruptions occurred from 1340 hours on 12 July, 1998 were observed using a
These data are available on Moodle in the “Datasets” folder called kiama.txt.
There is one variable called Interval which represents the “Waiting time between
(a) Does there appear to be a trend over time in the waiting times between eruptions?
i. Use RStudio to produce a suitable graphical display to show the pattern of
the successive times between eruptions of the blow hole over time.
ii. Comment on whether there appears to be a trend in waiting times over time.
(b) Now, considering this to be a sample of waiting times, this part of the question
asks you to explore the features of the distribution of waiting time.
i. Use RStudio to produce a boxplot and a frequency histogram of the distribution
of the set of waiting times.
ii. Comment on the major features of the distribution of the waiting times.
iii. Using your histogram or boxplot does it look plausible that the Kiama waiting
time data come from a normal distribution? Explain briefly why (or why
iv. Using the RStudio qqnorm() function, construct a normal quantile plot of
the Kiama data.
v. Explain how any skewness or symmetry of the data observed in the frequency
histogram and boxplot can be observed from the normal quantile plot.
vi. From this normal quantile plot, does it look plausible that the data comes
from a normal distribution? Explain briefly why (or why not).
(c) Next, transform the data by using a square root transformation and explore the
effect on the skewness. Note that if you have a vector of data called dat, say,
then sqrt(dat) will give you the vector of square roots.
i. Use RStudio to construct a boxplot and a frequency histogram of the transformed
Interval waiting times.
ii. Comment on the distribution as shown in the boxplot and histogram. Compare
this with the distribution of the data before it was transformed.
2. Let X1, . . . , Xn be iid random variables with the following probability density function
fX(x; θ) = (θ + 1)xθ, 0 < x < 1, and θ > 1.
(a) Find the method of moments estimator θe of θ.
(b) Find the maximum likelihood estimator θb of θ.
(c) Determine the probability density function of Y = ln(X).
3. A random variable is said to have an Exponential distribution if its probability density
function is given by
fX(x; β) = 1
βex/β , x > 0.
(a) Compute the maximum likelihood estimator βb of β.
(b) Compute the Fisher Information of βb.
(c) Derive an 95% confidence interval for β.
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