Let $X$ be a random variable with pdf $f_X(x) = 6x-6x^2$ if $0<x<1$, and $0$ otherwise. Let $Y=3X^2-2X^3$. Show that is a uniform distribution $U(0,1)$.
My attempt:
Specifically, $X = g^{-1}(Y)$ is a well defined inverse on $(0,1)$. The Jacobian of this transformation is $\left|\frac{d}{dy}g^{-1}(y)\right|$ is what I am struggling to find. Can someone help me on how to go about this?
$$y = g(x)$$ By using inverse function theorem, $$\frac{d}{dy}g^{-1}(y)=\frac1{g'(x)}=\frac1{6x-6x^2},$$
hence $Y$ is uniform.