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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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Independence is equivalent to the measure on the product space being the product measure. This is actually just a restatement of the definitions: $\mu_{X_1,...,X_n}(E_1,..,E_n) = P((X_1,...,X_n) \in (E_1,...,E_n) = \prod P(X_i \in E_i) = \prod \mu_i (E_i) = \mu_1 \times ... \times \mu_n (E_1 \times ... \times E_n)$

This is sufficient for Fubini's Theorem (modulo existence of integrals).

edit: Looking back at this, I suppose I should have been clearer since I may have misinterpreted your question. This is unambiguously true for product measures and could be true depending on what you mean by $E_X$ and $E_Y$ more generally. Let's just move down to the absolutely continuous case to make this a bit clearer: In general, we have $f_{X,Y}(x,y) = f_{X|Y}(x|y) f_Y (y)$ (this does generalize beyond the absolutely continuous case, but the notation is cumbersome). If your question does not involve conditioning then find any random variable for which the conditional probability $f_{X|Y} (x|y)$ does not equal the marginal $f_X (x)$. This is how I interpreted your original question, but that is probably not what you meant.

In light of your comment I will give a standard counterexample. Let $B = 1$ or $0$ each with probability $\frac{1}{2}$ and $D = 1$ or $-1$ each with probability $\frac{1}{2}$. Then $A = BD$ is uncorrelated with $B$, but $E A^2 B$ = $\frac{1}{2} \neq E_A E_B A^2 B = E_A A^2 E_B B = \frac{1}{4}$ where I am interpreting $E_B$ to mean the integral with respect to the marginal distribution of $B$.

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    +1. Thanks! (1) What does "modulo existence of integrals" mean? (2) How about uncorrelatedness?2012-02-11
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    Fubini's theorem is only a theorem when the integrand is absolutely integrable or positive. I'm reasonably certain that the variables being uncorrelated is not enough (as I said above if I recall it isn't hard to construct a counterexample). I'll play with it a bit and see if I come up with one.2012-02-11
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    Thanks for the edit! (1) My questions do not involve conditioning, just joint distribution and marginal distribution. Is this how you interpreted my questions? (2) I wonder what "If your question does not involve conditioning then find any random variable for which the conditional probability fX|Y(x|y) does not equal the marginal fX(x)" is for?2012-02-11
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    Yes, that is how I interpreted it originally. I'll add a counterexample to that to my answer then. That comment was for a way to find a counterexample, but I'll give an easier construction.2012-02-11
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    Thanks! (1) In the counterexample, do you mean $E e^{AB} \neq E_A [E_B e^{AB}]$ instead? (2) Maybe you have explained, but what does "modulo existence of integrals" mean?2012-02-11
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    That counterexample is wrong (or rather trivially not true), sorry. I was rushing. This is why you need to condition: if we work with marginals then there is no dependence of A on B, so $E_A AB = B E_A A$. By "modulo existence of integrals" I mean that the statement of Fubini's theorem requires that the integrand be absolutely integrable or positive. If neither of those is true then there are cases (even in the usual real analytic context) where the integral over the product space is not the double integral (or where the double integrals are not equal).2012-02-11
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    I was trying to be a little cute with the last one, so I edited in an easy to compute example of what can go wrong. Hopefully I didn't do anything silly this time :)2012-02-11
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    Thanks! (1) Do you mean $B$ when writing $X$? (2) $A$ and $B$ are uncorrelated, implying $A^2$ and $B$ are as well?2012-02-11
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    (1)Yes :) I'll go back and edit again. I have no idea why but all variables become X whenever I write something like this up. (2)No, but your question is if $X$ and $Y$ are uncorrelated can we split the integral of $f(X,Y)$ for Borel $f$. In this case $f(X,Y) = X^2 Y$. If $A^2$ and $B$ were uncorrelated then the first integral would have to be zero.2012-02-11
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    Thanks! I am not sure if I understand. $A$ and $B$ are uncorrelated, so I though you would say $E_{A,B} AB \neq E_A E_B AB $, but you wrote $E_{A,B} A^2 B \neq E_A E_B A^2B $ instead. Do you mean that $A^2$ and $B$ are uncorrelated because $A$ and $B$ are uncorrelated? I am guessing you would like to give a counterexample to my question about unrelatedness, did you?2012-02-11
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    Notice that $f(A,B) = A^2B$ is a Borel measurable function of $(A,B)$. Your question was if $A$ and $B$ are uncorrelated is it true that $E_{A,B} f(A,B) = E_A E_B f(A,B)$. In this case that is the question does $E_{A,B} A^2B = E_A E_B A^2 B$, which is what I gave a counterexample to.2012-02-11
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    Thanks! I think I was a little delirious. Sorry.2012-02-11
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    No problem. I probably wasn't terribly helpful with tripping over myself a couple of times in coming up with the example. Hope that helps though.2012-02-11