Suppose X, Y, and Z are independent geometric random variables with parameter $ \theta $. Now suppose V=G(X,Y) and U=F(Z). It seems intuitive that V and U would also be independent. The variation in this question is that V is a function of two random variables.
1 Answer
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Since the determination of X and Y are both independent of the determination of Z, it follows that the determination of any bivariate function of X and Y is independent of the determination of any monovariate function of Z.
(The converse does not follow, unless the functions are invertable.)