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(13) Let a, b, c, d be constants. If all of the following covariances are well defined then show that Cov(aX + bY,c2Z + dW) = ac Cov(X, Z) + ad Cov(X,W) + bc Cov(Y, Z) + bd Cov(Y, W). The following outlines of the proofs are proposed, where U= aX + bY and V = cZ+ dW. (a) Cov(U + V) - Cov(U) + Cov(V), which gives the result. (b) Since Cov(ÚV) = Cov(U)Cov(V), the result follows by substituting U = aX+bY and V = cZ +dW. (c) All we need to do is use the definition of covariance, then multiply out the expressions inside the expectation operation, E(UV), and then use the linearity of expectations to get this result. (d) The result, as stated, is false. (e) None of the above (a) (b) (c) (d) (e) N/A (Select One)

(13) Let a, b, c, d be constants. If all of the following covariances are well defined then show that Cov(aX + bY,c2Z + dW) = ac Cov(X, Z) + ad Cov(X,W) + bc Cov(Y, Z) + bd Cov(Y, W). The following outlines of the proofs are proposed, where U= aX + bY and V = cZ+ dW. (a) Cov(U + V) - Cov(U) + Cov(V), which gives the result. (b) Since Cov(ÚV) = Cov(U)Cov(V), the result follows by substituting U = aX+bY and V = cZ +dW. (c) All we need to do is use the definition of covariance, then multiply out the expressions inside the expectation operation, E(UV), and then use the linearity of expectations to get this result. (d) The result, as stated, is false. (e) None of the above (a) (b) (c) (d) (e) N/A (Select One)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 52RE
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(13) Let a, b, c, d be constants. If all of the following covariances are well defined then show that Cov(aX + bY, cZ + dW) = ac Cov(X, Z) + ad Cov(X, W) + bc Cov(Y, Z) + bd Cov(Y, W). The following outlines of the proofs are proposed, where U = aX+ bY and V = cZ + dW. (a) Cov(U + V) = Cov(U) + Cov(V), which gives the result. (b) Since Cov(ÚV) = Cov(U)Cov(V), the result follows by substituting U = aX+ bY and V = cZ + dW. (c) All we need to do is use the definition of covariance, then multiply out the expressions inside the expectation operation, E(UV), and then use the linearity of expectations to get this result. (d) The result, as stated, is false. (e) None of the above (a) (b) (c) (d) (e) N/A (Select One)
Transcribed Image Text:(13) Let a, b, c, d be constants. If all of the following covariances are well defined then show that Cov(aX + bY, cZ + dW) = ac Cov(X, Z) + ad Cov(X, W) + bc Cov(Y, Z) + bd Cov(Y, W). The following outlines of the proofs are proposed, where U = aX+ bY and V = cZ + dW. (a) Cov(U + V) = Cov(U) + Cov(V), which gives the result. (b) Since Cov(ÚV) = Cov(U)Cov(V), the result follows by substituting U = aX+ bY and V = cZ + dW. (c) All we need to do is use the definition of covariance, then multiply out the expressions inside the expectation operation, E(UV), and then use the linearity of expectations to get this result. (d) The result, as stated, is false. (e) None of the above (a) (b) (c) (d) (e) N/A (Select One)
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