Covariance matrix for multivariate normal random variable

Marcelo Maxwell 2022-09-23 Answered
Suppose we have a multivariate normal random variable X = [ X 1 , X 2 , X 3 , X 4 ] . And here X 1 and X 4 are independent (not correlated). Also X 2 and X 4 are independent. But X 1 and X 2 are not independent. Assume that Y = [ Y 1 , Y 2 ] is defined by
Y 1 = X 1 + X 4 ,     Y 2 = X 2 X 4 .
If I know the covariance matrix of X, what would be the covariance matrix of Y?
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Answers (1)

Edward Chase
Answered 2022-09-24 Author has 10 answers
Step 1
You can assume w.l.o.g. that E [ X ] = 0 . Then E [ Y ] = 0 (variances/covariances are not dependent on means).
You need to compute V a r ( Y 1 ) , V a r ( Y 2 ) , E [ Y 1 Y 2 ] since the covariance matrix of Y is comprised of these three elements.
Since X 1 , X 4 are independent, V a r ( Y 1 ) = V a r ( X 1 ) + V a r ( X 4 ) which you should konw from the covariance matrix of X.
Similarly for V a r ( Y 2 ) .
Finally you can compute E [ Y 1 Y 2 ] = E [ X 1 X 2 X 4 2 ] = C o v ( X 1 , X 2 ) V a r ( X 4 ) which you should know from the covariance matrix of X
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