Given distributions of 2 indpendent variables, find the probabiliy distribution of a function of the 2 independent random variables?

Question

Given $f_{X}$ and $f_{Y}$ how do you find the distribution $f_{g(X,Y)}$ of a function $g : \mathbb{R}^{2} \rightarrow \mathbb{R}$?

Answer 1

$P(g(X,Y) \leq t)=\int\int I_{\{(x,y): g(x,y) \leq t\}} f_X(x)f_Y(y)dxdy$. That's all one can say in general. Evaluation of the integral depends on the form of $g$.