# Joint distribution for sums of normal variables

Given $$n$$ independent normally distributed variables $$\{\xi_k\}_{k=1}^n$$. Each $$\xi_k$$ has expectation $$a$$ and dispersion $$\sigma^2$$.

I need to find joint distribution for two variables: $$\eta=\sum_{k=1}^n\xi_k$$ and $$\zeta=\sum_{k=1}^m\xi_k$$, $$m.

By definition, joint distribution function for $$(\eta,\zeta)$$ is $$F(y,z)=\int_{\sum_{k=1}^nx_iwhere $$p(x)=\frac{1}{\sigma\sqrt{2Pi}}e^{-\frac{(x-a)^2}{2\sigma^2}}$$

The question is how to proceed further with it.

Let $$A=\sigma I_n$$ and $$B=\begin{bmatrix} \mathbf{1_m^{\top}} & \mathbf{1}_{n-m}^{\top} \\ \mathbf{1_m^{\top}} & \mathbf{0}_{n-m}^{\top} \end{bmatrix}.$$ Then $$\begin{bmatrix} \zeta \\ \eta \end{bmatrix}=B(AZ+a\mathbf{1}_n),$$ where $$Z\sim N(0,I_n)$$. Therefore, $$[\zeta^{\top}, \eta^{\top}]^{\top}$$ is jointly normal with mean $$B\times a\mathbf{1}_n$$ and variance $$BAA^{\top}B^{\top}$$.