# Thread: Multi variate normal

1. ## Multi variate normal

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

2. Originally Posted by sharpe

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 $X_1$ and $X_2$ are normal then the linear combination $Y_1 = X_1+X_2$ is also normal.

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

Consider these

$E(aX_1+bX_2) =aE(X_1)+bE(X_2)$

$V(aX_1) = a^2V(X_1)$

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

It follows in your case that $a=b=1$ then

$E(Y_1) = E(X_1)+E(X_2)$

and

$V(Y_1) = V(X_1)+V(X_2)$

3. $(Y_1,Y_2)=\left(\matrix{1&1&0,\cr
1&0&1\cr}\right)X$

where that's supposed to be a 2 by 3 matrix

4. 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!