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4. (20 points) Consider two jars, Jar M and Jar W. In Jar M, there are 3 balls numbered 0, 1, 2. In Jar W there are 3 balls numbered 1, 2, 3. A ball is drawn from Jar M, then a ball is drawn from Jar W. Define M as the number on the ball from Jar M and W the number on the ball drawn from Jar W. Let X =WM and Y = min {W, M2} (Minimum of W and M2). (a) Find the joint probability mass function of X and Y. (b) Are X and Y independent? Use the definition of independence to support your answer. (c) Compute the Corr( VX, 2 ) (ie, Correlation coefficient of VX and 2). (d) Compute the Cov(22+Y, 4VX) and Corr(22+Y, 4 VX) . (e) Find the probability mass function of X given Y = 1 (ie, p(xly = 1))