 # Let B = ((1),(2)) and C= ((2),(1)). Define X=B^T A,Y=C^T A. How do I find the covariance of X and Y? potrefilizx 2022-09-14 Answered
Let the bivariate random variable A=(A1,A2)T have a Gaussian distribution on R2 with zero mean and covariance matrix be given by
$\left(\begin{array}{cc}1& -0.4\\ -0.4& 1\end{array}\right)$
Let B=$\left(\begin{array}{c}1\\ 2\end{array}\right)$ and C=$\left(\begin{array}{c}2\\ 1\end{array}\right)$. Define $X={B}^{T}A,Y={C}^{T}A$.How do I find the covariance of X and Y?
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You have ${B}^{\mathrm{\top }}A={A}_{1}+2{A}_{2}$ and ${C}^{\mathrm{\top }}A=2{A}_{1}+{A}_{2}$. Then,
$\begin{array}{rcl}Cov\left(X,Y\right)& =& Cov\left({A}_{1}+2{A}_{2},2{A}_{1}+{A}_{2}\right)\\ & =& 2Cov\left({A}_{1},{A}_{1}\right)+Cov\left({A}_{1},{A}_{2}\right)+4Cov\left({A}_{1},{A}_{2}\right)+2Cov\left({A}_{2},{A}_{2}\right).\end{array}$

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