Joint distribution of difference of normals

Question

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?

Answer 1

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?

Answer 2

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:

$$ \text{var}(X_1 - X_2) = \text{var}(X_1) + \text{var}(X_2) - 2 \text{cov}(X_1, X_2) $$