1. Let X denote the change in the number of international confrontations into which a nation...

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1. Let X denote the change in the number of international confrontations into which a nation entered from the quarter century preceding World War II to the quarter century following World War IIl; let Y denote the change in the gold reserves of a nation over the same two time periods. Suppose (X,Y)~N2 (H, . (μ.Σ) where 6. E = 4 = C). Find: (a) E(Y|X) [6 marks) (b) E(X|Y) (5 marks] (c) Var(Y|X) [4 marks) (d) Var(X|Y) [4 marks] 2. Let X be distributed as N3 (u, E), where u' 2) and = (1 -1 4 -1 = 1 Which of the following random variables are independent? Motivate your answer. 1 (a) X1 and X2 [3 marks] (b) X1 and X3 [3 marks] (c) X2 and X3 (2 marks] (d) (X1, X3) and X2 [3 marks) (e) X1 and X1 + 3X2 – 2X3 [14 marks] 1. Let X denote the change in the number of international confrontations into which a nation entered from the quarter century preceding World War II to the quarter century following World War IIl; let Y denote the change in the gold reserves of a nation over the same two time periods. Suppose (X,Y)~N2 (H, . (μ.Σ) where 6. E = 4 = C). Find: (a) E(Y|X) [6 marks) (b) E(X|Y) (5 marks] (c) Var(Y|X) [4 marks) (d) Var(X|Y) [4 marks] 2. Let X be distributed as N3 (u, E), where u' 2) and = (1 -1 4 -1 = 1 Which of the following random variables are independent? Motivate your answer. 1 (a) X1 and X2 [3 marks] (b) X1 and X3 [3 marks] (c) X2 and X3 (2 marks] (d) (X1, X3) and X2 [3 marks) (e) X1 and X1 + 3X2 – 2X3 [14 marks]