Suppose we have $2$ Independent random variables $X$ AND $Y$. Let $f(X)$ and $g(Y)$ are functions of those $2$ random variables.
1.) my question can we say that the functions $g(X)$ AND $f(Y)$ are independent as well?
2.) and what would be $\mathbb{E}[g(X)f(Y)]$ ?
The problem is that i m having that i m not sure whether $g(X)$ AND $f(Y)$ are independent if $X$ and $Y$ are independent.
The two random variables are indipendent.
Intuitively if x doesn't influence the value of y, so the image of x under f doesn't influence the other variable.
If the two variables are independent then E[g(X)*f(Y)] = E[g(X)]*E[f(X)] which is essentially the fubini theorem (remember the the definition of expectation).