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?
Best Answer
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.