# Multi variate normal

• Aug 7th 2010, 11:49 AM
sharpe
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
• Aug 7th 2010, 01:43 PM
pickslides
Quote:

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 $\displaystyle X_1$ 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

$\displaystyle V(Y_1) = V(X_1)+V(X_2)$
• Aug 7th 2010, 09:18 PM
matheagle
$\displaystyle (Y_1,Y_2)=\left(\matrix{1&1&0,\cr 1&0&1\cr}\right)X$
where that's supposed to be a 2 by 3 matrix
• Aug 8th 2010, 11:39 AM
sharpe