I have a problem where I have an object's PDF in position and velocity and I need it in position and time.
(It would be sufficient to just find $p(x | t)$, where $x$ is position and $t$ is time).
The distribution is normal, so let $\mathbf{x} \sim N(\mathbf{\mu}, \Sigma)$ where $\mathbf{x} = [x, v]^T$. Knowing $\mu$ and $\Sigma$ I already know $p(x, v)$, and it's pretty trivial to get $p(x | v)$.
I want to assert that $t = -\frac{x}{v}$ and find $p(x, t)$ -- but I'm hitting a wall finding that, or $p(x | t)$.