1. A test consists of 10 multiple choice (MC) questions, each of which has 5 choices. A student who hasn't studied decides to take the test and answer each question at random (i.e., pick one of the choices with equal probability). The test outcome can be repre- sented as a sequence of 10 Right or Wrong answers; e.g., (W, R, W, W, W, W, W, W, R, R) indicates that the student's answer to question I was wrong, to question 2 was right, to question 3 was wrong, and so on. (a) (2 points) Let the random variable (RV) X represent the total number of right answers. Name the distribution of X and specify the values of its parameters. (b) (2 points) Calculate the probability that the student gets 8 or more right answers. (c) (4 points) Let the RV Y be the question number of the first right answer. E.g., Y(W, W, W, R, W, W, R, W. R, W) = 4, because questions 1-3 were wrong and the 4th question was right. If the student didn't get any answers right, define the value of the RV Y to be 0, i.e. Y (W, . . ., W) = 0. Find the probability P(Y = 4) (d) (4 points) Derive the probability mass function (PMF) of Y, and verify that it is a valid PMF (i.e., show that the sum of all possible probabilities is 1). (e) (4 points) Now assume that you know that the student got 3 out of 10 questions right, but you don't know which ones (i.e., any 3 out of the 10 could be the right ones, with equal probability). Find the probability that exactly 2 out the first 5 questions were right, given that the student scored 3/10. (Hint: use the Hypergeometric distribution.) 2. (4 points) Assume that Toronto is hit by 3.5 snowstorms per year on average, where the actual number of snowstorms follows a Poisson distribution. Find the probability that there will be 4 or more snowstorms in Toronto next year