Question 10 Let S = {1, 2, 3, 4, 5, 6, 7, 8} be a sample space of an experiment, and assume that each element of S is equally likely to occur. Define the events: A1 = {1, 3, 5, 7}, A2 = {1, 2, 3, 4} and A3 = {6, 7,8}. (a) Give the probabilities of each of the events A1, A2 and A3. (b) Which of the following are true? Give reasons for your answers. (i) A1 and A2 are independent. (ii) A1 and A3 are independent. (iii) A2 and A3 are independent. (c) Define a random variable X by saying that if the observed outcome of the experiment is s, then the value of X is (s - 4)2. (i) Tabulate the probability mass function of the random variable X. (ii) Give the mean and variance of the random variable X