Thread: Multivariate Normal Distribution

I have taken 8 different books out of my library and still can't figure out how to do this question

Suppose X is distributed N2(μ, Σ). Determine the distribution of the random vector (X1 + X2, X1 - X2). Show that X1 + X2 and X1 - X2 are independent if Var(X1)=Var(X2)

Thanks a lot in advance!

You only need a calc3 book.
You can probably do this with MGFs, but it's easy enough to write out the joint density of X1 and X2.
Then obtain the jacobian and substitute W=X1+X2 and Z=X1-X2 into this new density.
Use the bivariate density in http://en.wikipedia.org/wiki/Multiva...l_distribution
and plug away