Joint distribution of difference of normals Consider the following three independent random variables, $X_1 \sim N(\mu_1,\sigma^2_1)$, $X_2 \sim N(\mu_2,\sigma^2_2)$, and $X_3 \sim N(\mu_3,\sigma^2_3)$.
Now let me define the pairwise differences as $\delta_{12} = X_1-X_2$, $\delta_{13} = X_1-X_3$, and $\delta_{23} = X_2-X_3$.
I know that these pairwise differences are easy to calculate as normal random variables, e.g.,
$$\delta_{ij}\sim N(\mu_i - \mu_j, \sigma^2_i + \sigma^2_j)$$
but are we able to calculate the joint distribution of $\delta_{12},\delta_{13},\delta_{23}$ in closed form?
In other words, what is the joint distribution of $p(\delta_{12},\delta_{13},\delta_{23}| \mu_1, \mu_2,\mu_3,\sigma^2_1,\sigma_2^2,\sigma_3^2)$?
Or do we need to place more assumptions?
 A: Write
\begin{align}
\Delta :=
\begin{bmatrix}
\delta_{12} \\
\delta_{13} \\
\delta_{23} 
\end{bmatrix} = 
\begin{bmatrix}
X_1 - X_2 \\
X_1 - X_3 \\
X_2 - X_3 
\end{bmatrix}
= \begin{bmatrix}
1 & -1 & 0  \\
1 & 0  & -1 \\
0 & 1  & -1 
\end{bmatrix}
\begin{bmatrix}
X_1 \\
X_2 \\
X_3 
\end{bmatrix} =: AX, 
\end{align}
then $\Delta \sim N_3(AE(X), A\operatorname{Cov}(X)A^T)$ by Affine transformation property of MVN.  Given that
\begin{align}
E(X) = 
\begin{bmatrix}
\mu_1 \\
\mu_2 \\
\mu_3 
\end{bmatrix}, \quad 
\operatorname{Cov}(X) 
= \begin{bmatrix}
\sigma_1^2 & 0 & 0  \\
0 & \sigma_2^2  & 0 \\
0 & 0  & \sigma_3^2 
\end{bmatrix},
\end{align}
can you finish the calculation?
A: It's trivial. The marginal means and variances remain the same, i.e. exactly as you have computed them. All you need to know is to calculate the off-diagonal of the variance-covariance matrix. To do so, this simple (useful) formula tells you what you need to know:
$$ \text{var}(X_1 - X_2) = \text{var}(X_1) + \text{var}(X_2) - 2 \text{cov}(X_1, X_2) $$
