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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Learning Outcomes:

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)

Answered 1 days AfterMar 16, 2022

SOLUTION.PDF## Answer To This Question Is Available To Download

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