Question
Download Solution PDFLet the correlation coefficient between X and Y be 0.6. Random variables Z and W are defined as Z = X + 5 and
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept
Let Z = aX + bY.
Expectation of Z = E(Z) = aE(X) + bE(Y)
Variance of Z = Var(Z) = a2Var(X) + b2Var(Y) + 2abρXYσXσY
ρXY = Correlation coefficient of X and Y.
Covariance of X and Y = Cov(X, Y) = E(XY) - E(X)E(Y)
If Z = X + a where a is some constant, then Var(Z) = Var(X)
Covariance of X and Y = Cov(X, Y) = E(XY) - E(X)E(Y)
Calculation
Z = X + 5
⇒ Var(Z) = Var(X)
⇒ σZ = σX
Cov(Z, W)
= E(ZW) – E(Z)E(W)
Correlation coefficient of Z and W is
= ρXY
So ρZW = ρXY = 0.6
Correct option is (4).
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