MATH2801代写、代做R程序设计、代写Moodle留学生、R设计代做

日期：2019-07-29 11:13

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

assignment.

Length: No more than 6 pages, including this cover sheet.

Name:

Student number:

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;

and/or,

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

plagiarism checking).

I (We) certify that I (We) have read and understood the University Rules in respect of

Student Academic Misconduct.

Signed: Date:

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

digital watch.

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

eruptions (sec)”.

(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

not).

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.