Density Transformation Theorem

probability-theory

Definition

Density Transformation Theorem

Let $X,Y$ be two random variables and $f_X,f_Y$ be two densities. Further let $g:\mathbb{R}\to \mathbb{R}$ be an injective function.

If $g$ has a differentiable inverse $g^{-1}$, then:

$$f_Y(y)=f_X(g^{-1}(y))\cdot \left|\frac{d}{dy}{g}^{-1}(y)\right|$$