# Multi variate normal

#### sharpe

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
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
$$\displaystyle (Y_1,Y_2)=\left(\matrix{1&1&0,\cr 1&0&1\cr}\right)X$$
where that's supposed to be a 2 by 3 matrix

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