The joint PDF of $X$ and the square of $X$ exists?

Question

In general how can you find the joint pdf of $X$ and $X^2$ if it exists? Can $f(x,x^2) = f(x^2)$, under the assumption that both $X$ and $X^2$ are Gaussian?

Answer 1

$(X,X^2)$ doesn't have a (joint) density w.r.t. the Lebesgue measure on $\mathbb{R}^2$ ($\because$ the distribution of $(X,X^2)$ is concentrated on the graph of $x\mapsto x^2$).