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Math Help - random variables separable into 2 functions integrated over the same variables

  1. #1
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    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.
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  2. #2
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    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.
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