Z is a normal, since this is linear in X and Y.

Thus just find the mean and variance of Z and you're done.

You can use MGFs, but that silly.

It does prove what I'm saying though.

Oh gave you the marginal of Z.

If you want the joint of X and Z, you need to transform, calc 3 from X,Y to X,Z.

Obtain the jacobian and substitute from one density to the other.