1

It's a bit messy, which might explain why it isn't seen more. Here is a sketch starting from the bivariate case, which generalizes. I'll use $X$ and $Y$ and $Z=X+Y.$ First let's find the conditional cdf for $Z$ given $X=x.$ $$F_{Z|X=x}=P \left[ X+Y \leq z \ | \ X=x\right]=P[Y \leq z-x]=F_Y(z-x)$$ Then the conditional pdf is found by differentiating: $$f_{...


1

It all follows from the properties of multivariate normals. Since $X_i$ are independent and normally distributed, they're jointly normal, which means any of their linear combination is also jointly normal with them. So, $p_{\mathbf{X},Z_N}(\mathbf{x},z)$ is a multivariate normal, which in turn means $p_{X_i,Z_N}(x,z)$ is multivariate normal with $$\mu=\...


Only top voted, non community-wiki answers of a minimum length are eligible