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(13) Let a, b, c, d be constants. If all of the followving 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(UV) = 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 followving 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(UV) = 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)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 32EQ
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(13) Let a, b, c, d be constants. If all of the followving 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(UV) = 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 followving 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(UV) = 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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