So we all know that if a probability density function of random variables X and Y is separable into the product of 2 functions f(x) and g(y), then X and Y are independent. However, if the probability density function of random variables X and Y is

are x and y necessarily independent? In other words, they are each being integrated over the same variables. I suspect they are NOT necessarily independent. Please prove or provide a counterexample if at all possible.