How to obtain a pdf of a random variable defined as a function of many variables?

Question

Given $N$ independent random variables ($X_1$,$X_2$,...,$X_N$) with individual pdfs $f_1$,...,$f_N$:

How to determine the pdf of a random variable $Y=G(X_1,...,X_N)$?

Answer 1

Well, the CDF would be $$ F_Y(y) = \mathbb{P}[Y \leq y] = \int_{A_y} \prod_{k=1}^N f_k(x_k) dx_1 \ldots dx_N, $$ where $A_y$ is the region of $\mathbb R^N$ where $G(x_1, \ldots, x_N) \leq y$ and the pdf is given by $$ f_Y(y) = \frac{dF_Y(y)}{dy}. $$