Problem 3: Joint PMF I have an four sided dice labeled 1,2,3,3 (not a typo two of the faces are three) that I roll once. Let X be the random variable that represents the result of the roll. I then flip a biased coin three times with probability of heads of % where :1: is the result of the roll. Let Y be the number of heads I ip. (1) Find the joint PMF of X, Y (ii) Find the marginal PMFs of X and Y. (iii) Find E[X + Y] (iv) Are X and Y independent. (v) Let A be the event that I ip zero heads. Find PX|A- (The PMF of X conditioned on A) (vi) Find E[X2 . Y|X > 1] (vii) Which value of w maximizes Var(Y|X = m)