Assume that X and are independent random variableswith: E(X) = 0, Var(X) = 4; E(

Question

Assume that X and are independent random variableswith: E(X) = 0,   Var(X) = 4;    E()= 0,   Var() = 2 ASSUME: Y= 3X + 2 + Then, using the properties of expectations andvariances, a) Find E(Y) and Var(Y) b) Find the covariance between X and Y: Cov (X,Y) = E(XY)- E(X) E(Y) Please SHOW ALL WORK. Thank you. Assume that X and are independent random variableswith: E(X) = 0,   Var(X) = 4;    E()= 0,   Var() = 2 ASSUME: Y= 3X + 2 + Then, using the properties of expectations andvariances, a) Find E(Y) and Var(Y) b) Find the covariance between X and Y: Cov (X,Y) = E(XY)- E(X) E(Y) Please SHOW ALL WORK. Thank you.

Explanation / Answer

Let a and b be 2 constants and assume X and Y are independent.Then, E(aX+b) = E(aX) + b = aE(X) + b and V(aX+Y) = V(aX) + V(Y) = a^2V(X) + V(Y) a) E(Y) = E(3X + 2 + )= E(3X) + 2 +E() = 3E(X) + 2 +0 = 3(0) + 2 = 2 Var(Y) = Var(3X + 2 + ) = Var(3X+) = Var(3X) +Var() = 3^2Var(X) + 2 = 9(4) + 2 = 38 b) Cov(X,Y) = Cov(X, 3X + 2 + )                 =Cov(X, 3X+)                = Cov(X, 3X) + Cov(X,)                = Cov(X, 3X)        Cov(X,) = 0 since X and are independent                = 3Cov(X,X)                = 3Var(X)                = 3(4)                = 12

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