Joint distribution of multivariate normal distribution So the question asks: Let $X = (X_1, ... ,X_{2n})$~ $ N (0, ∑)$ (multivariate normal distribution with mean vector $(0,..., 0)$ and covariance matrix $∑$ ), where $n≥ 1$. Find the joint distribution of the vector $(X_1 +...    + X_n,X_{n+1} + ...   + X_{2n})~ $.
So so far I got: 
The random vector $X = (X_1, ... ,X_{2n})$ will have a multivariate Gaussian distribution if the joint distribution of $X_1, ... ,X_{2n}$ has density
$\begin{align}f_X(X_1, ... ,X_{2n}) =& \dfrac 1{(2π) ^{2n/2} \det(Σ)^{1/2}}\exp(-(1 /2) (x − µ)^ tΣ^ {−1} (x − µ)) \\ =& \dfrac 1{(2π) ^{n} \det(Σ)^{1/2}}\exp(-(1 /2) (x − µ)^ tΣ^ {−1} (x − µ))\end{align}$
Is this right? But how do I suppose to do to find the joint distribution of the vector $(X_1 +...    + X_n,X_{n+1} + ...   + X_{2n})$?

Write 
\begin{align}
\begin{pmatrix}
Y_1 \\
Y_2
\end{pmatrix} :=
\begin{pmatrix}
X_1 + \cdots + X_n \\
X_{n + 1} + \cdots + X_{2n}
\end{pmatrix} =
\begin{pmatrix}
1 & \ldots & 1 & 0 & \ldots & 0 \\
0 & \ldots & 0 & 1 & \ldots & 1
\end{pmatrix}X := AX.
\end{align}
Then use if $X \sim N(\mu, \Sigma)$, then $AX \sim N(A\mu, A\Sigma A^T)$. So the remaining task is to find $A\Sigma A^T$, to simplify the result, you may partition $\Sigma$ according to the blocking structure of $A$ as
$$\Sigma = \begin{pmatrix}
\Sigma_{11} & \Sigma_{12} \\
\Sigma_{21} & \Sigma_{22}
\end{pmatrix}.$$
As a note, for problems of the same type, usually you don't need to deal with the density function directly, the invariance property of normal random vector under linear transformation in general would suffice. 
