Problem 1 (20+20 pts). i) Assume that three events A, B and C are independent. Show that A, B and C' are ndependent. 'ii) Let 0 = {1, 2, ...,60} be a sample space with the probability rule P(A) = #A/60, for A c Q. Define three random variables X, Y and Z by 1, if w is divisible by 3, X (W) = (1, if w is even, Y(w) = 10. otherwise; 10, if w is odd; Z ( W ) = 1, if w is divisible by 5, 0, otherwise. Show that X, Y and Z are independent. Problem 2 (20+20 pts). Assume that X and Y are independent and have the following distributions X 0 2 4 6 Y -101 Probabilities | 4 Probabilities 3 3 3 i) Calculate the mean values and the variances of X and Y. (ii) Calculate the distribution of Z = X + 2Y. Problem 3 (20 pts)