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08/11/2022, 18:55 Quiz: Quiz 1
Quiz 1 This is a preview of the published version of the quiz. Started: Nov 8 at 18:55 Quiz Instructions Question 1 1 pts Let be identically and independently distributed random variables, each with mean 0, variance 1, and moment generating function . Define the following quantities . Which of the following statement is not correct? Question 2 1 pts 0 Consider the two random variables and that have a bivariate normal distribution with mean standard deviation and correlation . What is the value of (hint: using conditional expectation)? 08/11/2022, 18:55 Quiz: Quiz 1 https://canvas.sydney.edu.au/courses/44955/quizzes/202277/take?preview=1 2/5 1 Question 3 1 pts p 0 words Consider tossing a six-sided die 20 times, and suppose the die is unbalanced so that the probability of observing an in each toss is . Let be the number of times appears out of 20 tosses. What is the distribution of ? You can just give the distribution without showing any work. (Hint: using intuition) Question 4 1 pts Consider the two discrete random variables and , each of which only receives a value of 1 or -1. Suppose the marginal distribution of is given by The conditional expectation 08/11/2022, 18:55 Quiz: Quiz 1 https://canvas.sydney.edu.au/courses/44955/quizzes/202277/take?preview=1 3/5 . What is the value for the marginal expectation ? For questions 5-8, please show all the workings. Upload your work to the corresponding Assignment item. Make sure you upload your work by 3:05pm (Wednesday 31 Aug – Sydney time). Question 5 (1.5 marks) Let be a continuous uniform random variable on . Find the density for . Make sure to specify the range of values (i.e the support) of . Question 6 (1.5 marks) Consider tossing a coin times; for each time, the probability that a coin lands head is . The number of heads, , is recorded. Then, the same coin is tossed for more times. What is the expectation for the total number of heads in this process? Question 7 ( 3 marks) 08/11/2022, 18:55 Quiz: Quiz 1 https://canvas.sydney.edu.au/courses/44955/quizzes/202277/take?preview=1 4/5 Saved at 18:55 Suppose the bivariate random variables and have the joint density Let . Find the joint density of and and show that and are independent. Question 8 (2 marks) Consider the hierarchical model . (a) Show that the (marginal) moment generating function of is given by You may find the expression useful for the proof. (b) Using the fact that prove that as , the random variable . Some useful formulas: Useful formulas.pdf (https://canvas.sydney.edu.au/courses/44955/files/26270143/download?download_frd=1) Submit Quiz 08/11/2022, 18:55 Quiz: Quiz 1
