Question 1: Let X and Y be independent normal random variables with distributions N(2, 9) and N(2, 4], respectively. (a) Find lE(2Y X). (b) Find the…

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Question 1: Let X and Y be independent normal randomvariables with distributions N(—2, 9) and N(2, 4], respectively. (a) Find lE(2Y — X).(b) Find the standard deviation ngwx. (cl Find W21” > X + Tl.: Let X1, X2 and X3 be mutually independent Question 2 : Let Var(X) : 9, Var(Y) : 25 and Var(X+Y) : 10 (we do not assume independence here). Find Var(2X _ 31’ + 4). Question 3: Let X1, X2 and X3 be mutually independentrandom variables, all with distribution U(0,1). Define two other random variables byU:X1 —4X2, V:2X2—X3. (a) Find ]E(U) and ]E(V). (b) Find Var(U) and Var(V). (c) Find Cov(U,V). (d) Find p(U, V). Comment on the sign and strength ofthc correlation.

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