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
( 1 0.4 0.4 1 )
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?
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Answers (1)

Jaden Mason
Answered 2022-09-15 Author has 15 answers
You have B A = A 1 + 2 A 2 and C A = 2 A 1 + A 2 . Then,
C o v ( X , Y ) = C o v ( A 1 + 2 A 2 , 2 A 1 + A 2 ) = 2 C o v ( A 1 , A 1 ) + C o v ( A 1 , A 2 ) + 4 C o v ( A 1 , A 2 ) + 2 C o v ( A 2 , A 2 ) .

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