MATH XXXXXXXXXX/22) Computational Methods
and Numerical Techniques
Contribution: 25% of
course
Course Leader:
Dr Kayvan Nejabati
Zenouz
Coursework 2 Deadline Date: Thursday
17/03/2022
This coursework will be marked anonymously
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4. Demonstrate knowledge of some of the commonly used statistical
techniques (simple linear regression), ca
y out the required statistical analysis and
eflect on results.
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MATH1180 COMPUTATIONAL METHODS
AND NUMERICAL TECHNIQUES
MATH1180 Coursework 2
Kayvan Nejabati Zenouz∗
15th Feb, 2022
Contents
Question 1: Probability of License Plates (20 marks) . . 4
Question 2: Roll a Die Twice (30 marks XXXXXXXXXX
Question 3: Continuous Random Variable (20 marks) . . 4
Question 4: Expectation and Variance (10 marks XXXXXXXXXX
Question 5: Joint Distribution (20 marks XXXXXXXXXX
Assignment Specification
• The coursework will be marked anonymously. Do not indicate your name.
Method marks may be awarded for partially completed solutions.
• You are required to explain each step of your solutions carefully and present
your work clearly in order to avoid losing marks.
• Some questions may require you to conduct research and use the resources
suggested in the reading list.
∗Office QM315, School of Computing and Mathematical Sciences, University of Greenwich,
Old Royal Naval College, London SE10 9LS, Email: XXXXXXXXXX, Student
Drop-in Hours: TUESDAYS 15:00-16:00 (QM315/TEAMS)
MATH1180 COURSEWORK 2 4
Question 1: Probability of License Plates (20 marks)
In some states, license plates have 5 characters: three letters followed by 2 numbers.
1. How many distinct such plates are possible? (5 marks)
2. If all sequences of five characters are equally likely, what is the probability that
the license plate for a new car will contain no duplicate letters or numbers?
(5 marks)
3. What is the probability that a randomly selected license plate contains both
A and 1? (5 marks)
4. What is the probability that a randomly selected license plate contains both
A and 1 given we know that the license plate contains no duplicate letters o
numbers? (5 marks)
Total: 20 marks
Question 2: Roll a Die Twice (30 marks)
Roll a fair dice twice. Let X be a discrete random variable representing the numbe
of the first roll minus the number on the second roll; for example, if first roll is 1
and second roll 2, then X(1, 2) = 1− 2 = −1.
1. Write down the sample space of the random variable X. (5 marks)
2. Create in a table for the probability mass function p(x) of X. (5 marks)
3. Find the cumulative distribution function CDF, F (x), for X and plot it against
the values of x. (10 marks)
4. Calculate the expectation E(X). (5 marks)
5. Calculate the variance Var(X). (5 marks)
Total: 30 marks
Question 3: Continuous Random Variable (20 marks)
Let X be a continuous random variable and suppose it has probability density
function of the form
f(x) =
{
α(1− x)n−1 for 0 ≤ x ≤ 1
0 otherwise,
where n is a known integer and n > 0.
Kayvan Nejabati Zenouz
XXXXXXXXXX
15th Feb, 2022 at 16:21
MATH1180 COURSEWORK 2 5
1. Find the value for α so that f(x) is in fact a probability density function.
(5 marks)
2. Derive the co
esponding cumulative distribution function, F (x). (5 marks)
3. Find a mathematical expression for the 0.25 quantile of X. That is find m so
that F (m) = XXXXXXXXXXmarks)
4. Assuming that n = 5, calculate P (0.5 < X < XXXXXXXXXXmarks)
Total: 20 marks
Question 4: Expectation and Variance (10 marks)
Let X be a continuous random variable and Y = aX + b for some constants a and
prove, using the definition for expectation and variance, the following.
1. E(Y ) = aE(X) + b. (5 marks)
2. Var(Y ) = a2Var(X). (5 marks)
Total: 10 marks
Question 5: Joint Distribution (20 marks)
Let X, Y have the joint PMF as shown in the following table.
y
p(x, y XXXXXXXXXX
XXXXXXXXXX03
x XXXXXXXXXX01
XXXXXXXXXX08
1. Find the marginal PMF for X and Y . (8 marks)
2. Find P (0.5 < X ≤ XXXXXXXXXXmarks)
3. Find P (Y = 1 | X = XXXXXXXXXXmarks)
4. Find Cov(X, Y ) covariance for X and Y . (8 marks)
Total: 20 marks
End of Assignment
Kayvan Nejabati Zenouz
XXXXXXXXXX
15th Feb, 2022 at 16:21
Question 1: Probability of License Plates 1mm(20 marks)
Question 2: Roll a Die Twice 1mm(30 marks)
Question 3: Continuous Random Variable 1mm(20 marks)
Question 4: Expectation and Variance 1mm(10 marks)
Question 5: Joint Distribution 1mm(20 marks)