Let X and Y be jointly normal, with the mean vectors and covariance matrix below:
Let
Find the mean vector and covariance matrix of
I figured out the following:
But how do I figure out ?
Hello,
Well, the elements of the covariance matrix are the covariances...
So we have cov(X,Y)=cov(Y,X)=.5
And finding cov(Z1,Z2) is then easy, using the fact that the covariance is a bilinear form (cov(ax,y)=a*cov(x,y) , cov(x+y,z)=cov(x,z)+cov(y,z))
Otherwise, you can note that :
Let's denote
And we know that will follow a normal distribution
See a proof in the attached pdf