(4) Suppose you roll a pair of fair dice, each with sides labeled 1, 2, 3, 4, 5, 6. Let X denote the sum of the numbers showing on the two dice. Let Y denote the larger of the two numbers showing on the two dice. (i.e. If the roll is (3, 5), then X = 8 and Y 5. If the roll is (4,4), then X 8 and Y = 4.) (a) Describe the distribution of X. It might help to draw a 6 x 6 array to think about S and X. (b) Describe the distribution of Y. It might help to draw a 6 × 6 array to think about S and X. (c) Compute P(X = 7) and P(X > 7). (d) Compute P(Y = 3) and P(Y < 3). (e) Compute P(X = 7|Y = 3) and P(X = 7|Y = 6). %3D (f) Are the events {X = 7} and {Y = 6} independent?

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