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

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