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 $\zeta=\sum_{k=1}^m\xi_k$, $m<n$.

By definition, joint distribution function for $(\eta,\zeta)$ is $$F(y,z)=\int_{\sum_{k=1}^nx_i<y,\sum_{k=1}^mx_i<z}\prod_{k=1}^np(x_i)dx_1dx_2...dx_n$$where $$p(x)=\frac{1}{\sigma\sqrt{2Pi}}e^{-\frac{(x-a)^2}{2\sigma^2}}$$

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 $BAA^{\top}B^{\top}$.


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