Show that two continuous random variables x and y are independent (i.e. p(x, y) = p(x )p(y) for all x and y) if and only if their joint distribution function F(x, y) satisfies F(x, y) = F(x) F(y) for all x and y. Prove that the same thing is true for discrete random variables. [This is an example of a result which is easier to prove in the continuous case.]