Transformation of Random Variables: How to relate these two approaches?

Question

In the literature, I have found two approaches for the transformation of random variables

$$p_{X}(x) =\int_{-\infty}^{\infty}p_{Y}(y)\delta\left(x-f(y)\right)~\mathrm{d}y$$

and

$$p_{X}(x)=p_{Y}(f^{-1}(x))\left|\frac{\mathrm{d}f^{-1}(x)}{\mathrm{d}y}\right|.$$

How do you manage to reconcile these two?

My feeble attempt

$$p_{X}(x) =\int_{-\infty}^{\infty}p_{Y}(y)\delta\left(x-f(y)\right)~\mathrm{d}y=\int_{-\infty}^{\infty}p_{Y}(y)\delta\left(y-f^{-1}(x)\right)~\mathrm{d}y=p_{Y}\left(f^{-1}(x)\right)$$

is missing a term. Sorry my foolish question and feeble attempt but it really bothers me.

Answer 1

$\newcommand{\dd}{\mathop{}\!\mathrm{d}}$ By properties of the Dirac $\delta$ and the inverse function theorem, we have $$\delta(x-f(y))=\frac{\delta\left(y-f^{-1}(x)\right)}{\lvert f'(f^{-1}(x))\rvert}=\delta\left(y-f^{-1}(x)\right)\cdot\left\lvert(f^{-1})'(x)\right\rvert.$$ The above is essentially applying a "change of variables" type formula to the Dirac $\delta$. It obviously generalises the scaling and shifting properties. See e.g. this math.SE thread.

So plugging this form in, $$p_X(x)=\int_{\infty}^\infty p_Y(y)\delta\left(y-f^{-1}(x)\right)\cdot\left\lvert(f^{-1})'(x)\right\rvert\dd y=p_Y\left(f^{-1}(x)\right)\cdot\left\lvert(f^{-1})'(x)\right\rvert,$$ which is as desired.