Find the conditional PDF of a multivariate normal distribution given a constraint

Question

Problem to solve

We have a vector of random variables $\textbf{X}=(X_1,X_2)$ issued from a bivariate normal distribution. In particular, $\mu = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$, $\Sigma = \begin{bmatrix} 2 & 1\\ 1 & 2\end{bmatrix}$. Find the conditional PDF, $f(X_1|X_1 + X_2 = 5)$.

Approach so far

Main objective: Looking for the ingredients to compute $f(X_1|X_1 + X_2 = 5) = \frac{f(X_1,X_1 + X_2 = 5)}{f(X_1 + X_2 = 5)}$.

First step: find $f(X_1 + X_2)$.

This PDF should follow a normal distribution due to the properties of normal distributions, $\textbf{AX}\sim\mathcal{N}(\textbf{A}\mu, \textbf{A}\Sigma\textbf{A}')$, for $\textbf{A}$ a matrix of finite real constants. Then,

$\mathbb{E}[X_1 + X_2] = \mathbb{E}[X_1] + \mathbb{E}[X_2] = 5$,

$\mathbb{V}[X_1 + X_2] = \mathbb{V}[X_1] + \mathbb{V}[X_2] + 2\mathbb{C}\textrm{ov}[X_1, X_2]=6$.

Therefore, $X_1 + X_2 \sim \mathcal{N}(5, 6)$, and,

$f(X_1+X_2=5) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left(-\frac{1}{2\sigma^2}(X_1 + X_2 -\mu)^2\right)}$,

$\Rightarrow f(X_1+X_2=5) = \frac{1}{2\sqrt{3\pi}}$,

Second step: find $f(X_1,X_1 + X_2)$.

Again, this joint PDF should follow a multivariate normal distribution due to the properties of normal distributions.

Consider a linear transformation $Z = X_1 + X_2$, then,

$f_{X_1,X_1 + X_2}(X_1, X_1 + X_2) = f_{X_1, Z}(X_1, Z)\frac{\partial Z}{\partial X_1}=f_{X_1, Z}(X_1, X_1 + X_2)$.

And now... See next section.

Alternative approach (considering comments):

Set $A = \begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$ in $\textbf{AX}\sim\mathcal{N}(\textbf{A}\mu, \textbf{A}\Sigma\textbf{A}')$. Then, it follows,

$\begin{bmatrix}X_1 \\ X_1 + X_2\end{bmatrix} \sim \mathcal{N}\left(\begin{bmatrix}2 \\ 5\end{bmatrix}, \begin{bmatrix}2 & 3 \\ 3 & 6\end{bmatrix}\right)$.

This way, we get the joint distribution $f(X_1,X_1 + X_2)$ and its marginal distribution $f(X_1 + X_2)$ simply and directly.

Now we can evaluate $f(X_1 + X_2)$ when $X_1 + X_2 = 5$ like before and get,

$f_{X_1 + X_2}(5) = \frac{1}{2\sqrt{3\pi}}$.

Now we look for, $f(X_1, X_1+X_2 = 5)$. Confused about the procedure now.

Issue

I am not sure on how to go forward.

1. Should $f(X_1, Z)$ be evaluated directly, if yes, how so? How can I find the mean and the variance covariance matrix of this PDF?
2. If not, how do I find $f(X_1, X_1 + X_2)$?
3. Else, is the overall approach wrong to find $f(X_1|X_1 + X_2 = 5)$?

Connected topics

Here is a list of connected topics:

1. Conditional Distribution Multivariate Normal Distribution [closed]