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Suppose we have two random variables $X, Y$. Then, in general, if they are dependent $$E[XY] \ne E[X]E[Y]$$

However, according to the law of total expectation, $$E[XY] = E_Y[E_X[XY|Y]] = E_Y[YE[X]]=E[Y]E[X].$$ Furthermore, this law doesn't stipulate that $X$ and $Y$ are independent.

How can we reconcile this?

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  • $\begingroup$ Ah that makes sense, if you make it an answer I can accept it $\endgroup$ Mar 8, 2021 at 18:19

1 Answer 1

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The law of total expectation states that $$\mathbb E_X[X]=\mathbb E_Y[\mathbb E_{X|Y}[X|Y]] $$ Introducing $Y$ leads to $$\mathbb E_{(X,Y)}[YX]=\mathbb E_Y[\mathbb E_{X|Y}[XY|Y]] = \mathbb E_Y[Y\mathbb E_{X|Y}[X|Y]] $$ which differs from $\mathbb E_Y[Y]\mathbb E_{X}[X]$ when $\mathbb E_{X|Y}[X|Y]$ depends on $Y$.

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    $\begingroup$ Why is the first expectation $\mathbb{E}_X[YX]$ only over $X$? $\endgroup$
    – Kuku
    Jun 9, 2023 at 13:06
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    $\begingroup$ @Kuku: a typo, it is. And there was another one, with the $XY$ in the subscript rather than the integrand. Thank you. $\endgroup$
    – Xi'an
    Jun 9, 2023 at 14:58
  • $\begingroup$ Thank you for correcting. I am confused though still, shouldn't the second term have $XY|Y$ as subscript instead of $X|Y$? $\endgroup$
    – Kuku
    Jun 14, 2023 at 8:35
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    $\begingroup$ No, although it does not make a difference in the end. $\endgroup$
    – Xi'an
    Jun 14, 2023 at 9:45

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