# Transformation of Random Variable from $\mathbb R^n$ to $\mathbb R$

Say $$X\sim F$$ is a random vector is $$\mathbb R^n$$ with $$F\ll\lambda$$ (where $$\lambda$$ is the Lebesgue Measure in $$\mathbb R^n$$). I'm interested in the density of $$Y=X^\top X$$. How do I show that $$f_{Y}(y)=\int_{x^\top x=y} \frac{dF}{d\lambda}d\nu$$ where $$\nu$$ is the uniform distribution over the set $$\{x\in\mathbb R^n|x^\top x = y\}$$. Intuitively it makes perfect sense to me but I guess I'm having trouble taking the derivative of the distribution of $$Y$$. As in I know that $$\mathbb P(Y \le y) = \int_{x^\top x \le y} \frac{dF}{d\lambda}d\lambda$$ but it's not very clear to me how to obtain the first equation. It looks somewhat like Green's Theorem since $$\{x^\top x =y\}=\partial \{x^\top x \le y\}$$ but I'm not sure if I can use that.

• This is a consequence of the coarea formula.
– FZan
Commented Mar 7 at 23:58
• @FZan that solves it! Would you consider posting this as the solution? Commented Mar 9 at 2:49