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Question 5: Joint Distribution (20 marks) Let X, Y have the joint PMF as shown in the following table. p(x, y) 1 2 3 4 0.07 0.05 0.13 0.03 X 0.07 0.06 0.1 0.01 3 0.09 0.24 0.07 0.08 1. Find the marginal PMF for X and Y. (8 marks) p(x, y) = P(X( = x, Y = y). P( ISX YSI ) py (1) = p(1, D) + p( 2. 1) + p(3, 1) = 0.07 + 0.07 + 0.09 = 0.23 py (2) = p( 1, 2) + p( 2. 2) + p(3. 2) = 0.05 + 0.06 + 0.24 = 0.35 py (3) = p( 1. 3) + p( 2. 3) + p(3.3) = 0.13 + 0.1 + 0.07 = 0.3 py (3) = p( 1, 4) + p(2. 4) + p(3.4) = 0.03 + 0.01 + 0.08 = 0.12p(x, y) 1 2 3 4 PX(x) 1 0.07 0.05 0.13 0.03 0.28 0.07 0.06 0.1 0.01 0.24 3 0.09 0.24 0.07 0.08 0.48 PY(y) 0.23 0.35 0.3 0.12 2. Find P (0.5 XP(X=x) = 1(0.28) + 2(0.24) + 3(0.48) =2.2 E(n = _ YP( Y=Y) - 1(0.23) + 2(0.35) + 3(0.3) + 4(0.12) = 1.96 Cov (X. Y) - 5.04 - (2.2) (1.96) = 0.728