Thread: random variables separable into 2 functions integrated over the same variables

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.

Just about any reasonable hierarchical model with X and Y being conditionally independent (independent given Z, say) is going to show that this is false.

For the sake of concreteness, suppose is standard normal and and are conditionally independent and identically distributed given . It is easy to show . But the joint pdf of X and Y can be expressed in that form, by writing down the joint pdf of X, Y, and Z, and then integrating out Z.