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Suppose $X$ and $Y$ are two real-valued random variables, and $f:\mathbb{R}^2\to \mathbb{R}$ is Borel measurable.

I was wondering if $X$ and $Y$ being uncorrelated or independent implies that $$ \mathrm{E}_{X,Y} f(X,Y) = \mathrm{E}_X [\mathrm{E}_Y f(X,Y)] = \mathrm{E}_Y [\mathrm{E}_X f(X,Y)]? $$ Thanks and regards!

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    The hard part of answering your question will be figuring out what your notation means.2012-02-11
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    I might be abusing notations, but I am not sure. Could you let me know what you think?2012-02-11
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    I deleted the comment and made it an answer, sorry. The random variables being merely uncorrelated is not sufficient. It is not hard to construct a counterexample from whatever your favorite uncorrelated not-independent random variables are.2012-02-11
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    @Chris: Thanks! I was wondering what causes ambiguity to you and/or Michael in my notations?2012-02-13
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    Generally $E_x$ refers to the conditional expectation given that $X = x$ rather than the expectation with respect to the marginal law of $X$.2012-02-13
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    @Chris: Thanks! (1) Is $E_X$ generally used? If yes, what does it generally mean? (2) How would you rewrite $\mathrm{E}_{X,Y} f(X,Y) = \mathrm{E}_X [\mathrm{E}_Y f(X,Y)] = \mathrm{E}_Y [\mathrm{E}_X f(X,Y)]$ to mean the result of Fubini's theorem?2012-02-13
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    I honestly don't know the answer to either of those questions. Personally I would go back to the integral formulation for the latter though. Expectations are nice and all, but it's often confusing when it is unclear which measure you are supposed to integrate with respect to.2012-02-13
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    @MichaelHardy: I was wondering what causes ambiguity to you in my notations? If possible, could you also look at the above comments of mine and Chris related to this question? Thanks!2012-02-13

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