I’m currently working on a problem where I need to calculate the density of $(a,b)X=(aX,bX)$ where $a,b>0$ and $X$ has density $f$ .
However, I’m facing some difficulties as the Jacobian method, which I initially thought to use, is not applicable in my case. According to this source, the Jacobian method cannot be used as my function $h: R \to R^2$ is not applicable.
It’s worth noting that for the one-dimensional case, we have that the density of $aX$ is given by $\frac{1}{a}f(\frac{x}{a})$.
According to the comments below, $(aX,bX)$ has no density. I don't have much background in advanced probability. Could you help me understand the intuition of why density does not exist?
Any help would be greatly appreciated.
Thank you!