ELE2035 Mathematics and Algorithms - Coursework

This coursework covers the probability and statistics part of the ELE2035 Mathematics and
Algorithms module. Where applicable, all solutions must be obtained using MATLAB which is
available on the University open access computers and here.

Submission: Coursework solutions must be uploaded to Canvas by 5 pm on Tuesday 3rd May
2022. Penalties apply for late submission.

Please note the following:

1. This coursework is worth 20% of the overall module marks;

2. The total number of marks assigned to each question is indicated. In total, you have 5
questions to solve.

3. You should prepare a PDF or Word document with answers for each question.

For each question, you must provide the following information:

(a) Any results, explanation and figures that you have been asked for;
(b) Where applicable, a MATLAB script that can executed to provide the solution to
each part of the question.















Q1. The time to failure of an organic light-emitting diode (OLED) television
can be modelled using the Weibull distribution with a location
parameter of zero, = 800 and = 4.1.


Using MATLAB, determine the probability that the television will fail in
the first 730 hours.


[10 marks]


Q2. Download and import the attached MATLAB workspace file data.mat.

It contains an unknown set of data, x, that we would like to perform an
exploratory data analysis on.


In MATLAB, determine the mean, variance, interquartile range and
the coefficients of skewness and kurtosis of x.

Comment on the coefficients of skewness and kurtosis.



[20 marks]


Q3. In the workspace, there is a second set of data y.

Using MATLAB, compute the covariance and correlation coefficient
between the two data sets, x and y.

Provide a scatter plot with a line of best fit.

Comment on any relationship that may be apparent between x and y
based on the statistical tools used above.






[20 marks]


Q4. Next perform a two-tailed ‘z-test’ at a significance level of 0.05 to
ascertain whether or not y comes from a normal distribution with
mean 0 and a (known) standard deviation of 2.

As part of your answer provide your null and alternative hypotheses,
the calculated p-value and the outcome from the test as well as any
MATLAB code used to obtain the result.






[20 marks]


Q5. Let X and Y represent two standard normal random variables. Let R
represent a third random variable which is formed from X and Y as
follows:
= √2 + 2


In MATLAB, generate 10,000 realisations (i.e. random values) of R.

Following from this, plot the empirical CDF of the generated data.



A parsimonious model (not covered in class) is a model that achieves a
desired level of explanation using as few parameters as possible. Find
the simplest probability distribution that models the distribution of
R.

Plot the theoretical CDF of your chosen model on the same plot as the
empirical CDF and comment on the fit and choice for the model.





[30 marks]