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

1. ## 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

$\int^\infty_{-\infty} \int^\infty_{-\infty} .... \int^\infty_{-\infty} f(x,u_1,u_2,....u_n)*g(y,u_1,u_2,....u_n)\,du_1\,d u_2 .... \,du_n$

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.

2. 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 $Z$ is standard normal and $X$ and $Y$ are conditionally independent and identically distributed $N(z, 1)$ given $Z = z$. It is easy to show $\mbox {Cov}[X, Y] \ne 0$. 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.