Maths questionMATH1180 (2021/22) Computational Methods and Numerical Techniques Contribution: 25%...

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Maths questionMATH1180 (2021/22) 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  YOU MUST NOT PUT ANY INDICATION OF YOUR IDENTITY IN YOUR  SUBMISSION  This coursework should take an average student who is up-to-date with tutorial  work approximately 50 hours  Feedback and grades are normally made available within 15 working days of the  coursework deadline  Learning Outcomes:  4. Demonstrate knowledge of some of the commonly used statistical  techniques (simple linear regression), carry out the required statistical analysis and  reflect on results. Plagiarism is presenting somebody else's work as your own. 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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) . . . . . 4 Question 3: Continuous Random Variable (20 marks) . . 4 Question 4: Expectation and Variance (10 marks) . . . 5 Question 5: Joint Distribution (20 marks) . . . . . 5 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: K.NejabatiZenouz@greenwich.ac.uk, 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 or 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 number 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 K.NejabatiZenouz@greenwich.ac.uk 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 corresponding 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) = 0.25. (5 marks) 4. Assuming that n = 5, calculate P (0.5

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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...

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