It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed. If I have two independent random variables, X1 and X2, then I define two other random variables Y1 and Y2, where Y1 = f1(X1) and Y2 = f2(X2). Intuitively, Y1 and Y2 should be independent, and I can't find a counter example, but I am not sure. Could anyone tell me whether they are independent? Does it depend on some properties of f1 and f2?