1. The problem statement, all variables and given/known data Let X and Y be independent random variables. Prove that g(X) and h(Y) are also independent where g and h are functions. 2. Relevant equations I did some research and somehow stumbled upon how E(XY) = E(X)E(Y) is important in the proof. f(x,y) = f(x)f(y) F(x,y) = F(x)F(y) 3. The attempt at a solution I am seriously stuck on this - I do not even know where to begin. I know there was a previous thread (around 7 years old) on this but I did not want to revive an old thread, and there wasn't much response in that thread. I attempted to determine F(g(x),h(y)) by integration but then I would need f(g(x),h(y)), which (to my knowledge), is unknown unless I use a really messy transformation equation (and the question was stated prior to learning about transformations). Expectations appear to be the way to go, since they do not require the actual density/probability function of the function of the random variable; however, I do not know of any property of expectations allowing us to deduce that two random variables are independent (since a covariance of 0 does not necessarily imply independence). If anyone could provide me with somewhere to start, that would be much appreciated.