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This question is an aside from another question here on CV.

We know that the expectation of the product of two independent random variables is the product of expectations, i.e., $$\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$$

However, is there a conditionally equivalent version of this statement? For example, could I say if $X$ and $Y$ are dependent random variables then by conditional independence we have

$$\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y|X]$$

I know that we can factor joint distributions that way using conditional properties, for example, $$f(x,y)=f(x)f(y|x)$$ and so is there an equivalence for expectations?

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Your second equality is certainly untrue unless $X$ and $Y$ are independent.

An easy way to view it is that the left side of the equality is a number while the right side of the equality is in general a random variable. Keep in mind that the conditional expectations are random variables.

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A consequence of conditional independence is

$$E \left[ X Y | Z \right] = E \left[X|Z \right] E\left[Y|Z \right]$$

and the idea is often encountered in hierarchical models. Is that perhaps what you had in mind?

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  • $\begingroup$ Thanks for that but no. I guess I want to know if my second equality is true. $\endgroup$ – user95564 Dec 4 '15 at 0:53
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    $\begingroup$ @RustyStatistician You can get $E\left[XY \right] = E \left[ X E\left( Y | X \right) \right]$ simply by applying the Law of Iterated Expectations but I have not seen anywhere what you have written. $\endgroup$ – JohnK Dec 4 '15 at 0:55

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