Multi variate normal

Jul 2010
26
0
Suppose X = (x1, x2, x3)' is multivariate normal

((3 2 2), ((4 -2 -2) (-2 2 1) (-2 1 2)))

(for the 3*3 matrix, the first row is 4 -2 -2 and second -2 2 1 etc...)

What is the joint distribution function Y1 = X1 + X2 and Y2 = X1+X3?

multivariates are relatively new to me, could I ask for how this might be calculated?

Many thanks
 

pickslides

MHF Helper
Sep 2008
5,237
1,625
Melbourne
What is the joint distribution function Y1 = X1 + X2
Hi there sharpe, this is a good question.

What you need to know is that if \(\displaystyle X_1\) and \(\displaystyle X_2\) are normal then the linear combination \(\displaystyle Y_1 = X_1+X_2\) is also normal.

So you need to find the expectation and variance of this joint distribution.

Consider these

\(\displaystyle E(aX_1+bX_2) =aE(X_1)+bE(X_2)\)

\(\displaystyle V(aX_1) = a^2V(X_1)\)

\(\displaystyle V(aX_1+bX_2) = a^2V(X_1)+b^2V(X_2)+2ab\times COV(X_,X_2)\)

It follows in your case that \(\displaystyle a=b=1\) then

\(\displaystyle E(Y_1) = E(X_1)+E(X_2)\)

and

\(\displaystyle V(Y_1) = V(X_1)+V(X_2)\)
 

matheagle

MHF Hall of Honor
Feb 2009
2,763
1,146
\(\displaystyle (Y_1,Y_2)=\left(\matrix{1&1&0,\cr
1&0&1\cr}\right)X\)
where that's supposed to be a 2 by 3 matrix
 
Last edited:
Jul 2010
26
0
Thanks that is helpful.

For some reason I could not see what the second matrix represented, but with your comments the penny dropped and it shows the variances and covariances. It was then pretty trivial

Thanks a lot!