We know if two variables are functionally independent then there will be stochastic independence as well. Can any one give me an example when the converse is not true i.e. $X$ and $Y$ are independent (stochastic) but not functionally independent?
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4$\begingroup$ How is functionally independence between random variables defined? $\endgroup$– Stefan HansenCommented Jun 24, 2014 at 8:58
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1$\begingroup$ X is not a function of Y and Y is not a function of X as well. That is functional independence. @StefanHansen $\endgroup$– arnabCommented Jun 25, 2014 at 6:14
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1$\begingroup$ Take $X=Y=1$, for example. $\endgroup$– Stefan HansenCommented Jun 25, 2014 at 6:49
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We know if two variables are functionally independent then there will be stochastic independence as well.
This is wrong. If $X, Y$ have a bivariate normal distribution woth correlation $\rho=\frac12$, they are clearly functionally independent (there do not exist some function $f$ such that $f(X,Y)$ is almost surely a constant, but they are not independent. Even with the reformulation that if two variables are stochastically independent, they will be functionally independent you must be careful, it will not be true if they have zero variance, for instance (two almost surely constant random variables are independent!)
As for your question I have just given the example. Takeaway: You need to be more careful with your definitions.
answered Mar 30, 2019 at 22:16
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$\begingroup$ +1 for the interesting definition of "functionally independent". A reference for further reading would be welcome. $\endgroup$– GalenCommented Oct 21, 2021 at 20:25
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1$\begingroup$ @Galen: You can compare with the definition given here: physicsforums.com/threads/functional-independence.75805 Also math.stackexchange.com/questions/1110763/… $\endgroup$ Commented Oct 21, 2021 at 20:28