How do I intuitively understand the following result to find the probability density function $P_Y(y)$ given $P_X(x)$ after change of variables $y=f(x)$ or several variables. How to derive this from scratch?
$$P_Y(y)=\int{dx\delta(y-f(x))P_X(x)}$$ How to understand it's extension to a function $z=g(x_1,x_2...x_n)$ of several variables given their respective pdf.
$$P_Y(y)=\int{dx_1...dx_n\delta(z-f(x_1...x_n))P_{X_1}(x_1)...P_{X_n}(x_n)}$$
Consider that integrating against $\delta(y-f(x))$ will pick out $P_X(x)$ for all $x$ such that $f(x)=y$ for the given (fixed) $y$. It follows from substituting $y'=f(x)$ that (in the countable case) $$\begin{align}\int P_X(x)\delta(y-f(x))\, dx&=\int P_X(f^{-1}(y'))\delta(y-y')\, \frac1{|f'(f^{-1}(y'))|}dy'\\&=\sum_{x,\ f(x)=y} \frac1{|f'(x)|}P_X(x)\end{align}$$ i.e. $P_Y(y)$ is given by a sum of the contributions $P_X(x)$ for all the $x$ that map to $y$.
