Examination Period EXAM CODES: ECE2191 TITLE OF PAPER: Mock Exam EXAM DURATION: Rules During an exam, you must not have in your possession any item/material that has not been authorised for your exam. This includes books, notes, paper, electronic device/s, mobile phone, smart watch/device, calculator, pencil case, or writing on any part of your body. Any authorised items are listed below. Items/materials on your desk, chair, in your clothing or otherwise on your person will be deemed to be in your possession. You must not retain, copy, memorise or note down any exam content for personal use or to share with any other person by any means following your exam. You must comply with any instructions given to you by an exam supervisor. As a student, and under Monash University’s Student Academic Integrity procedure, you must undertake your in-semester tasks, and end-of-semester tasks, including exams, with honesty and integrity. In exams, you must not allow anyone else to do work for you and you must not do any work for others. You must not contact, or attempt to contact, another person in an attempt to gain unfair advantage during your exam session. Assessors may take reasonable steps to check that your work displays the expected standards of academic integrity. Failure to comply with the above instructions, or attempting to cheat or cheating in an exam may constitute a breach of instructions under regulation 23 of the Monash University (Academic Board) Regulations or may constitute an act of academic misconduct under Part 7 of the Monash University (Council) Regulations. Authorised Materials OPEN BOOK YES NO CALCULATORS YES NO Faculty Approved DICTIONARIES YES NO NOTES YES NO SPECIFICALLY PERMITTED ITEMS YES NO if yes, items permitted are: Instructions Instructions Page 1 of 8 This is an open-book exam. You are permitted to use lecture notes and calculators to complete the questions. This exam consists of 7 questions. You have 2 hours and 10 minutes to complete the exam. The exam has the following sections: Section A: Essay and Composite Questions Answer the question in section A by entering text online in the dedicated space below each question. Note that these questions may have multiple parts and all parts should be attempted. Section B: Hybrid Questions Answer all of the hybrid questions in section B on your own pieces of paper. All questions in section B should be attempted. Please clearly label each blank piece of paper with your Student ID and the question number (and subpart of the question, if applicable). Please do not write your name on the paper. You will have time at the end of your exam to upload photographs of your answer sheets. If you believe there is an error or a question is unclear, clearly state any assumptions you need to make and proceed. Page 2 of 8 Information This is an open-book exam. You are permitted to use lecture notes and calculators to complete the questions. This exam consists of 7 questions. You have 2 hours and 10 minutes to complete the exam. The exam has the following sections: Section A: Essay and Composite Questions Answer the question in section A by entering text online in the dedicated space below each question. Note that these questions may have multiple parts and all parts should be attempted. Section B: Hybrid Questions Answer all of the hybrid questions in section B on your own pieces of paper. All questions in section B should be attempted. Please clearly label each blank piece of paper with your Student ID and the question number (and subpart of the question, if applicable). Please do not write your name on the paper. You will have time at the end of your exam to upload photographs of your answer sheets. If you believe there is an error or a question is unclear, clearly state any assumptions you need to make and proceed. Page 3 of 8 Section A: Essay and Composite Questions Question 1 My partner and I are one of 10 married couples at a dinner party. The 20 people are given random seats around a round table. What is the probability that I am seated next to my spouse? Explain your results in full by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.) 4 Marks Question 2 The PDF of a continuous random variable \(X\) is shown below: The random variable \(Y\) is related to \(X\) as follows: Answer the following questions. Explain your results in full by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.) What is the possible value of a? (1 Mark) What is the probability of even X=1? (2 Mark) Find the probability mass function of Y. (4 Marks) 7 Marks Page 4 of 8 Section B: Hybrid Questions Question 3 In a binary transmission system, the transmitted signal \(X\) is either -1 or 1, depending on whether sending a “0” bit or a “1” bit, respectively. The received signal \(Y\) is corrupted by an additive noise \ (N\) with a zero-mean Gaussian distribution with variance \(\sigma ^2=1/16\), i.e. \(Y=X+N\). Assume that “0” bits are two times as likely as “1” bits. a) Assume that the receiver decides a “0” was transmitted if the observed value of y satisfies \(f_Y(y|X=-1)P(X=- 1)>f_Y(y|X=1)P(X=1)\) and it decides a “1” was transmitted otherwise. Find a threshold \(T\) where this decision rule becomes equivalent to deciding “0” if \(y
(4 Marks) b) What is the overall probability of error? (3 Marks) Explain your results in full by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.) You may use the following Q-function table: Please answer question on your blank piece of paper. After your exam finishes, you’ll have extra time to access your phone to scan a QR code and upload your answer. Clearly label each page with Student ID and this question number (and sub part if applicable) (for example, 'Question 7a') Do not write your Name on it No. of answer sheets: 4 7 Marks Page 5 of 8 Question 4 Anytime Fae throws a Frisbee, her dog catches the Frisbee with probability \(p=0.1\), independent of whether the Frisbee is caught on any previous throw. When the dog catches the Frisbee, it runs away with the Frisbee, never to be seen again. Fae continues to throw the Frisbee until the dog catches it. Let X denote the number of times the Frisbee is thrown. a) Find an expression for PMF of X? (3 marks) b) What is the probability that Fae will throw the Frisbee more than three times? (3 marks) c) Find the expected value of X. (3 marks) d) Now assume that her dog always returns the Frisbee after catching it, so that Fae can throw it again. Every time her dog returns with the Frisbee, Fae rewards her with a meatball. Find an expression for PMF of the number of Meatballs given to the dog if Fae throws the Frisbee 100 times. (3 marks) Please answer question on your blank piece of paper. After your exam finishes, you’ll have extra time to access your phone to scan a QR code and upload your answer. Clearly label each page with Student ID and this question number (and sub part if applicable) (for example, 'Question 7a') Do not write your Name on it No. of answer sheets: 1 12 Marks Question 5 Answer both of the following: (a) Let \(Z = aX + bY\), where \(a\) and \(b\) are constants, and \(X\) and \(Y\) are continuous random variables. Show that \(E(Z) = E(aX + bY) = aE(X) + bE(Y)\). (4 marks) (b) Let \(G = aF + b\), where \(a\) and \(b\) are constants, and \(F\) is a continuous random variable. Find the covariance of \(F\) and \(G\). (6 marks) Justify your answers by including all reasoning and/or formulae used to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.) Please answer question on your blank piece of paper. After your exam finishes, you’ll have extra time to access your phone to scan a QR code and upload your answer. Clearly label each page with Student ID and this question number (and sub part if applicable) (for example, 'Question 7a') Do not write your Name on it No. of answer sheets: 4 10 Marks Page 6 of 8 Question 6 Consider an experiment of drawing randomly three balls from a bowl containing two red, three white and four blue balls. Let (X,Y) be a pair of random variables, where X and Y denote, respectively, the number of red and white balls chosen. Answer all of the following parts to the question: (a) Find the joint probability mass function of (X,Y). (4 marks) (b) Find the marginal probability mass functions of X and Y. (3 marks) (c) Are X and Y independent? (3 marks) Justify your answers by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.) Please answer question on your blank piece of paper. After your exam finishes, you’ll have extra time to access your phone to scan a QR code and upload your answer. Clearly label each page with Student ID and this question number (and sub part if applicable) (for example, 'Question 7a') Do not write your Name on it No. of answer sheets: 4 10 Marks Page 7 of 8 Question 7 The diameter of a particular brand of cricket ball is approximately normally distributed, with a mean of 7.12cm and a standard deviation of 0.075cm. If you select a random sample of nine cricket balls: (a) What is the probability that the sample mean is less than or equal to 7.07cm? (3 marks) (b) What is the probability that the sample mean is between 7.095cm and 7.145cm? (3 marks) (c) The probability is 60% that the sample mean will be between what two values symmetrically distributed around the population mean? (4 marks) Answer all of the above parts to the question. Justify your answers by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.) Please answer question on your blank piece of paper. After your exam finishes, you’ll have extra time to access your phone to scan a QR code and upload your answer. Clearly label each page with Student ID and this question number (and sub part if applicable) (for example, 'Question 7a') Do not write your Name on it No. of answer sheets: 4 10 Marks Page 8 of 8 欢迎咨询51作业君
